{"id":1339,"date":"2026-03-06T12:15:43","date_gmt":"2026-03-06T10:15:43","guid":{"rendered":"https:\/\/blog.edu.turku.fi\/allu\/?page_id=1339"},"modified":"2026-03-25T14:46:27","modified_gmt":"2026-03-25T12:46:27","slug":"syt","status":"publish","type":"page","link":"https:\/\/blog.edu.turku.fi\/allu\/sivu-1\/lukuteoria\/syt\/","title":{"rendered":"Eukleideen algoritmi"},"content":{"rendered":"\n<div class=\"wp-block-buttons is-layout-flex wp-block-buttons-is-layout-flex\">\n<div class=\"wp-block-button\"><a class=\"wp-block-button__link has-black-color has-text-color has-background has-link-color wp-element-button\" href=\"https:\/\/blog.edu.turku.fi\/allu\/sivu-1\/lukuteoria\/\" style=\"background-color:#d6d6ff\">Palaa takaisin<\/a><\/div>\n<\/div>\n\n\n\n<div style=\"height:43px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">Eukleideen algoritmi<\/h2>\n\n\n\n<div style=\"height:43px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<p class=\"has-text-align-left\">Teht\u00e4v\u00e4t on jaoteltu v\u00e4rin mukaan helppoihin, perus ja vaikeisiin teht\u00e4viin.<\/p>\n\n\n\n<div class=\"wp-block-buttons is-layout-flex wp-block-buttons-is-layout-flex\">\n<div class=\"wp-block-button\"><a class=\"wp-block-button__link has-black-color has-text-color has-background has-link-color wp-element-button\" style=\"background-color:#e6f3b7\">Helppo<\/a><\/div>\n\n\n\n<div class=\"wp-block-button\"><a class=\"wp-block-button__link has-black-color has-text-color has-background has-link-color has-text-align-center wp-element-button\" style=\"background-color:#d6d6ff\">Perus<\/a><\/div>\n\n\n\n<div class=\"wp-block-button\"><a class=\"wp-block-button__link has-black-color has-text-color has-background has-link-color wp-element-button\" style=\"background-color:#fcd8c8\">Vaikea<\/a><\/div>\n<\/div>\n\n\n\n<div style=\"height:43px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\" style=\"font-size:18px\"><strong>Teht\u00e4v\u00e4 1.1.<\/strong> Kirjoita jakoyht\u00e4l\u00f6 seuraaville jakolaskuille.<\/h3>\n\n\n\n<p><strong>a)<\/strong> <math data-latex=\"\\frac {21}{6}\"><semantics><mfrac><mn>21<\/mn><mn>6<\/mn><\/mfrac><annotation encoding=\"application\/x-tex\">\\frac {21}{6}<\/annotation><\/semantics><\/math><br><br><strong>b)<\/strong> <math data-latex=\"\\frac{128}{23}\"><semantics><mfrac><mn>128<\/mn><mn>23<\/mn><\/mfrac><annotation encoding=\"application\/x-tex\">\\frac{128}{23}<\/annotation><\/semantics><\/math><br><br><strong>c)<\/strong> <math data-latex=\"\\frac{48}{10}\"><semantics><mfrac><mn>48<\/mn><mn>10<\/mn><\/mfrac><annotation encoding=\"application\/x-tex\">\\frac{48}{10}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<div class=\"wp-block-group is-layout-constrained wp-block-group-is-layout-constrained\">\n<div data-wp-context=\"{ &quot;autoclose&quot;: false, &quot;accordionItems&quot;: [] }\" data-wp-interactive=\"core\/accordion\" role=\"group\" class=\"wp-block-accordion is-layout-flow wp-block-accordion-is-layout-flow\">\n<div data-wp-class--is-open=\"state.isOpen\" data-wp-context=\"{ &quot;id&quot;: &quot;accordion-item-1&quot;, &quot;openByDefault&quot;: false }\" data-wp-init=\"callbacks.initAccordionItems\" data-wp-on-window--hashchange=\"callbacks.hashChange\" class=\"wp-block-accordion-item is-layout-flow wp-block-accordion-item-is-layout-flow\">\n<h3 class=\"wp-block-accordion-heading has-background has-medium-font-size\" style=\"background-color:#e6f3b7\"><button aria-expanded=\"false\" aria-controls=\"accordion-item-1-panel\" data-wp-bind--aria-expanded=\"state.isOpen\" data-wp-on--click=\"actions.toggle\" data-wp-on--keydown=\"actions.handleKeyDown\" id=\"accordion-item-1\" type=\"button\" class=\"wp-block-accordion-heading__toggle\"><span class=\"wp-block-accordion-heading__toggle-title\">VINKKI!<\/span><span class=\"wp-block-accordion-heading__toggle-icon\" aria-hidden=\"true\">+<\/span><\/button><\/h3>\n\n\n\n<div inert aria-labelledby=\"accordion-item-1\" data-wp-bind--inert=\"!state.isOpen\" id=\"accordion-item-1-panel\" role=\"region\" class=\"wp-block-accordion-panel is-layout-flow wp-block-accordion-panel-is-layout-flow\">\n<p>Jakoyht\u00e4l\u00f6 on muotoa <math data-latex=\"a=b*q+r\"><semantics><mrow><mi>a<\/mi><mo>=<\/mo><mi>b<\/mi><mo>\u2217<\/mo><mi>q<\/mi><mo>+<\/mo><mi>r<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a=b*q+r<\/annotation><\/semantics><\/math> eli <math data-latex=\"jaettava = jakaja \\cdot  osam\u00e4\u00e4r\u00e4 + jakoj\u00e4\u00e4nn\u00f6s\"><semantics><mrow><mi>j<\/mi><mi>a<\/mi><mi>e<\/mi><mi>t<\/mi><mi>t<\/mi><mi>a<\/mi><mi>v<\/mi><mi>a<\/mi><mo>=<\/mo><mi>j<\/mi><mi>a<\/mi><mi>k<\/mi><mi>a<\/mi><mi>j<\/mi><mi>a<\/mi><mo>\u22c5<\/mo><mi>o<\/mi><mi>s<\/mi><mi>a<\/mi><mi>m<\/mi><mover><mi>a<\/mi><mo stretchy=\"false\" class=\"tml-xshift\">\u00a8<\/mo><\/mover><mover><mi>a<\/mi><mo stretchy=\"false\" class=\"tml-xshift\">\u00a8<\/mo><\/mover><mi>r<\/mi><mover><mi>a<\/mi><mo stretchy=\"false\" class=\"tml-xshift\">\u00a8<\/mo><\/mover><mo>+<\/mo><mi>j<\/mi><mi>a<\/mi><mi>k<\/mi><mi>o<\/mi><mi>j<\/mi><mover><mi>a<\/mi><mo stretchy=\"false\" class=\"tml-xshift\">\u00a8<\/mo><\/mover><mover><mi>a<\/mi><mo stretchy=\"false\" class=\"tml-xshift\">\u00a8<\/mo><\/mover><mi>n<\/mi><mi>n<\/mi><mover><mi>o<\/mi><mo stretchy=\"false\" class=\"tml-xshift\">\u00a8<\/mo><\/mover><mi>s<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">jaettava = jakaja \\cdot  osam\u00e4\u00e4r\u00e4 + jakoj\u00e4\u00e4nn\u00f6s<\/annotation><\/semantics><\/math><\/p>\n<\/div>\n<\/div>\n<\/div>\n\n\n\n<div data-wp-context=\"{ &quot;autoclose&quot;: false, &quot;accordionItems&quot;: [] }\" data-wp-interactive=\"core\/accordion\" role=\"group\" class=\"wp-block-accordion is-layout-flow wp-block-accordion-is-layout-flow\">\n<div data-wp-class--is-open=\"state.isOpen\" data-wp-context=\"{ &quot;id&quot;: &quot;accordion-item-2&quot;, &quot;openByDefault&quot;: false }\" data-wp-init=\"callbacks.initAccordionItems\" data-wp-on-window--hashchange=\"callbacks.hashChange\" class=\"wp-block-accordion-item is-layout-flow wp-block-accordion-item-is-layout-flow\">\n<h3 class=\"wp-block-accordion-heading has-background has-medium-font-size\" style=\"background-color:#e6f3b7\"><button aria-expanded=\"false\" aria-controls=\"accordion-item-2-panel\" data-wp-bind--aria-expanded=\"state.isOpen\" data-wp-on--click=\"actions.toggle\" data-wp-on--keydown=\"actions.handleKeyDown\" id=\"accordion-item-2\" type=\"button\" class=\"wp-block-accordion-heading__toggle\"><span class=\"wp-block-accordion-heading__toggle-title\">Vastaus<\/span><span class=\"wp-block-accordion-heading__toggle-icon\" aria-hidden=\"true\">+<\/span><\/button><\/h3>\n\n\n\n<div inert aria-labelledby=\"accordion-item-2\" data-wp-bind--inert=\"!state.isOpen\" id=\"accordion-item-2-panel\" role=\"region\" class=\"wp-block-accordion-panel is-layout-flow wp-block-accordion-panel-is-layout-flow\">\n<p>a) <math data-latex=\"21 = 6 \\cdot 3+3\"><semantics><mrow><mn>21<\/mn><mo>=<\/mo><mn>6<\/mn><mo>\u22c5<\/mo><mn>3<\/mn><mo>+<\/mo><mn>3<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">21 = 6 \\cdot 3+3<\/annotation><\/semantics><\/math><br>b) <math data-latex=\"128=23\\cdot5+13\"><semantics><mrow><mn>128<\/mn><mo>=<\/mo><mn>23<\/mn><mo>\u22c5<\/mo><mn>5<\/mn><mo>+<\/mo><mn>13<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">128=23\\cdot5+13<\/annotation><\/semantics><\/math><br>c) <math data-latex=\"48=10\\cdot 4+8\"><semantics><mrow><mn>48<\/mn><mo>=<\/mo><mn>10<\/mn><mo>\u22c5<\/mo><mn>4<\/mn><mo>+<\/mo><mn>8<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">48=10\\cdot 4+8<\/annotation><\/semantics><\/math><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n\n\n\n<div style=\"height:43px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\" style=\"font-size:18px\"><strong>Teht\u00e4v\u00e4 1.2. <\/strong>Mitk\u00e4 ovat kohdan 1.1. jakoyht\u00e4l\u00f6iden kokonaisosat ja jakoj\u00e4\u00e4nn\u00f6kset?<\/h3>\n\n\n\n<div data-wp-context=\"{ &quot;autoclose&quot;: false, &quot;accordionItems&quot;: [] }\" data-wp-interactive=\"core\/accordion\" role=\"group\" class=\"wp-block-accordion is-layout-flow wp-block-accordion-is-layout-flow\">\n<div data-wp-class--is-open=\"state.isOpen\" data-wp-context=\"{ &quot;id&quot;: &quot;accordion-item-3&quot;, &quot;openByDefault&quot;: false }\" data-wp-init=\"callbacks.initAccordionItems\" data-wp-on-window--hashchange=\"callbacks.hashChange\" class=\"wp-block-accordion-item is-layout-flow wp-block-accordion-item-is-layout-flow\">\n<h3 class=\"wp-block-accordion-heading has-background has-medium-font-size\" style=\"background-color:#e6f3b7\"><button aria-expanded=\"false\" aria-controls=\"accordion-item-3-panel\" data-wp-bind--aria-expanded=\"state.isOpen\" data-wp-on--click=\"actions.toggle\" data-wp-on--keydown=\"actions.handleKeyDown\" id=\"accordion-item-3\" type=\"button\" class=\"wp-block-accordion-heading__toggle\"><span class=\"wp-block-accordion-heading__toggle-title\">Vastaus<\/span><span class=\"wp-block-accordion-heading__toggle-icon\" aria-hidden=\"true\">+<\/span><\/button><\/h3>\n\n\n\n<div inert aria-labelledby=\"accordion-item-3\" data-wp-bind--inert=\"!state.isOpen\" id=\"accordion-item-3-panel\" role=\"region\" class=\"wp-block-accordion-panel is-layout-flow wp-block-accordion-panel-is-layout-flow\">\n<p>a) kokonaisosa q=3 ja jakoj\u00e4\u00e4nn\u00f6s r=3<br>b) kokonaisosa q=5 ja jakoj\u00e4\u00e4nn\u00f6s r=13<br>c) kokonaisosa q=4 ja jakoj\u00e4\u00e4nn\u00f6s r=8<\/p>\n<\/div>\n<\/div>\n<\/div>\n\n\n\n<div style=\"height:43px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\" style=\"font-size:18px\"><strong>Teht\u00e4v\u00e4 2.<\/strong> Selvit\u00e4 suurin yhteinen tekij\u00e4 ilman laskutoimituksia, tarkastelemalla annettuja v\u00e4livaihteita.<\/h3>\n\n\n\n<p><strong>a)<\/strong> syt(99, 63)<\/p>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>v\u00e4livaiheet<\/summary>\n<p><math data-latex=\"99=63\\cdot1+36\"><semantics><mrow><mn>99<\/mn><mo>=<\/mo><mn>63<\/mn><mo>\u22c5<\/mo><mn>1<\/mn><mo>+<\/mo><mn>36<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">99=63\\cdot1+36<\/annotation><\/semantics><\/math><br><math data-latex=\"63=36\\cdot1+27\"><semantics><mrow><mn>63<\/mn><mo>=<\/mo><mn>36<\/mn><mo>\u22c5<\/mo><mn>1<\/mn><mo>+<\/mo><mn>27<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">63=36\\cdot1+27<\/annotation><\/semantics><\/math><br><math data-latex=\"36=27\\cdot1+9\"><semantics><mrow><mn>36<\/mn><mo>=<\/mo><mn>27<\/mn><mo>\u22c5<\/mo><mn>1<\/mn><mo>+<\/mo><mn>9<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">36=27\\cdot1+9<\/annotation><\/semantics><\/math><br><math data-latex=\"27=9\\cdot3+0\"><semantics><mrow><mn>27<\/mn><mo>=<\/mo><mn>9<\/mn><mo>\u22c5<\/mo><mn>3<\/mn><mo>+<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">27=9\\cdot3+0<\/annotation><\/semantics><\/math><\/p>\n<\/details>\n\n\n\n<div style=\"height:25px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<p><strong>b)<\/strong> syt(187, 121)<\/p>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>v\u00e4livaiheet<\/summary>\n<p>a<math data-latex=\"187=121\\cdot1+66\"><semantics><mrow><mn>187<\/mn><mo>=<\/mo><mn>121<\/mn><mo>\u22c5<\/mo><mn>1<\/mn><mo>+<\/mo><mn>66<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">187=121\\cdot1+66<\/annotation><\/semantics><\/math><br><math data-latex=\"121=66\\cdot1+55\"><semantics><mrow><mn>121<\/mn><mo>=<\/mo><mn>66<\/mn><mo>\u22c5<\/mo><mn>1<\/mn><mo>+<\/mo><mn>55<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">121=66\\cdot1+55<\/annotation><\/semantics><\/math><br><br><math data-latex=\"55=11\\cdot 5+0\"><semantics><mrow><mn>55<\/mn><mo>=<\/mo><mn>11<\/mn><mo>\u22c5<\/mo><mn>5<\/mn><mo>+<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">55=11\\cdot 5+0<\/annotation><\/semantics><\/math><\/p>\n<\/details>\n\n\n\n<div style=\"height:25px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<p><strong>c)<\/strong> syt(143,104)<\/p>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>v\u00e4livaiheet<\/summary>\n<p><math data-latex=\"143=104\\cdot1+39\"><semantics><mrow><mn>143<\/mn><mo>=<\/mo><mn>104<\/mn><mo>\u22c5<\/mo><mn>1<\/mn><mo>+<\/mo><mn>39<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">143=104\\cdot1+39<\/annotation><\/semantics><\/math><br><math data-latex=\"104=39\\cdot2+26\"><semantics><mrow><mn>104<\/mn><mo>=<\/mo><mn>39<\/mn><mo>\u22c5<\/mo><mn>2<\/mn><mo>+<\/mo><mn>26<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">104=39\\cdot2+26<\/annotation><\/semantics><\/math><br><math data-latex=\"39=26\\cdot1+x\"><semantics><mrow><mn>39<\/mn><mo>=<\/mo><mn>26<\/mn><mo>\u22c5<\/mo><mn>1<\/mn><mo>+<\/mo><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">39=26\\cdot1+x<\/annotation><\/semantics><\/math><br><math data-latex=\"26=x\\cdot2+0\"><semantics><mrow><mn>26<\/mn><mo>=<\/mo><mi>x<\/mi><mo>\u22c5<\/mo><mn>2<\/mn><mo>+<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">26=x\\cdot2+0<\/annotation><\/semantics><\/math><\/p>\n<\/details>\n\n\n\n<div style=\"height:25px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<div data-wp-context=\"{ &quot;autoclose&quot;: false, &quot;accordionItems&quot;: [] }\" data-wp-interactive=\"core\/accordion\" role=\"group\" class=\"wp-block-accordion is-layout-flow wp-block-accordion-is-layout-flow\">\n<div data-wp-class--is-open=\"state.isOpen\" data-wp-context=\"{ &quot;id&quot;: &quot;accordion-item-4&quot;, &quot;openByDefault&quot;: false }\" data-wp-init=\"callbacks.initAccordionItems\" data-wp-on-window--hashchange=\"callbacks.hashChange\" class=\"wp-block-accordion-item is-layout-flow wp-block-accordion-item-is-layout-flow\">\n<h3 class=\"wp-block-accordion-heading has-background has-medium-font-size\" style=\"background-color:#e6f3b7\"><button aria-expanded=\"false\" aria-controls=\"accordion-item-4-panel\" data-wp-bind--aria-expanded=\"state.isOpen\" data-wp-on--click=\"actions.toggle\" data-wp-on--keydown=\"actions.handleKeyDown\" id=\"accordion-item-4\" type=\"button\" class=\"wp-block-accordion-heading__toggle\"><span class=\"wp-block-accordion-heading__toggle-title\">Vastaus<\/span><span class=\"wp-block-accordion-heading__toggle-icon\" aria-hidden=\"true\">+<\/span><\/button><\/h3>\n\n\n\n<div inert aria-labelledby=\"accordion-item-4\" data-wp-bind--inert=\"!state.isOpen\" id=\"accordion-item-4-panel\" role=\"region\" class=\"wp-block-accordion-panel is-layout-flow wp-block-accordion-panel-is-layout-flow\">\n<p>a) Suurin yhteinen tekij\u00e4 on 9<\/p>\n\n\n\n<p>b) Suurin yhteinen tekij\u00e4 on 11<\/p>\n\n\n\n<p>c) Suurin yhteinen tekij\u00e4 on 13<\/p>\n<\/div>\n<\/div>\n<\/div>\n\n\n\n<div style=\"height:43px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\" style=\"font-size:18px\"><strong>Teht\u00e4v\u00e4 3.<\/strong> Laske suurin yhteinen tekij\u00e4 Eukleideen algoritmin avulla.<\/h3>\n\n\n\n<p><strong>a)<\/strong> syt(32, 18)<br><strong>b)<\/strong> syt(244, 56)<br><strong>c)<\/strong> syt(721, 679)<\/p>\n\n\n\n<div data-wp-context=\"{ &quot;autoclose&quot;: false, &quot;accordionItems&quot;: [] }\" data-wp-interactive=\"core\/accordion\" role=\"group\" class=\"wp-block-accordion is-layout-flow wp-block-accordion-is-layout-flow\">\n<div data-wp-class--is-open=\"state.isOpen\" data-wp-context=\"{ &quot;id&quot;: &quot;accordion-item-5&quot;, &quot;openByDefault&quot;: false }\" data-wp-init=\"callbacks.initAccordionItems\" data-wp-on-window--hashchange=\"callbacks.hashChange\" class=\"wp-block-accordion-item is-layout-flow wp-block-accordion-item-is-layout-flow\">\n<h3 class=\"wp-block-accordion-heading has-background has-medium-font-size\" style=\"background-color:#d6d6ff\"><button aria-expanded=\"false\" aria-controls=\"accordion-item-5-panel\" data-wp-bind--aria-expanded=\"state.isOpen\" data-wp-on--click=\"actions.toggle\" data-wp-on--keydown=\"actions.handleKeyDown\" id=\"accordion-item-5\" type=\"button\" class=\"wp-block-accordion-heading__toggle\"><span class=\"wp-block-accordion-heading__toggle-title\">Vastaus<\/span><span class=\"wp-block-accordion-heading__toggle-icon\" aria-hidden=\"true\">+<\/span><\/button><\/h3>\n\n\n\n<div inert aria-labelledby=\"accordion-item-5\" data-wp-bind--inert=\"!state.isOpen\" id=\"accordion-item-5-panel\" role=\"region\" class=\"wp-block-accordion-panel is-layout-flow wp-block-accordion-panel-is-layout-flow\">\n<p>a) syt(32, 18)<br><math data-latex=\"32=18\\cdot1+14\"><semantics><mrow><mn>32<\/mn><mo>=<\/mo><mn>18<\/mn><mo>\u22c5<\/mo><mn>1<\/mn><mo>+<\/mo><mn>14<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">32=18\\cdot1+14<\/annotation><\/semantics><\/math><br><math data-latex=\"18=14\\cdot1+4\"><semantics><mrow><mn>18<\/mn><mo>=<\/mo><mn>14<\/mn><mo>\u22c5<\/mo><mn>1<\/mn><mo>+<\/mo><mn>4<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">18=14\\cdot1+4<\/annotation><\/semantics><\/math><br><math data-latex=\"14=4\\cdot3+2\"><semantics><mrow><mn>14<\/mn><mo>=<\/mo><mn>4<\/mn><mo>\u22c5<\/mo><mn>3<\/mn><mo>+<\/mo><mn>2<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">14=4\\cdot3+2<\/annotation><\/semantics><\/math><br><math data-latex=\"4=2\\cdot 2+0\"><semantics><mrow><mn>4<\/mn><mo>=<\/mo><mn>2<\/mn><mo>\u22c5<\/mo><mn>2<\/mn><mo>+<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">4=2\\cdot 2+0<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p>vastaus: Suurin yhteinen tekij\u00e4 on 2<\/p>\n\n\n\n<p>b) syt(244,56)<br><math data-latex=\"244=56\\cdot 4+20\"><semantics><mrow><mn>244<\/mn><mo>=<\/mo><mn>56<\/mn><mo>\u22c5<\/mo><mn>4<\/mn><mo>+<\/mo><mn>20<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">244=56\\cdot 4+20<\/annotation><\/semantics><\/math><br><math data-latex=\"56=20\\cdot2+16\"><semantics><mrow><mn>56<\/mn><mo>=<\/mo><mn>20<\/mn><mo>\u22c5<\/mo><mn>2<\/mn><mo>+<\/mo><mn>16<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">56=20\\cdot2+16<\/annotation><\/semantics><\/math><br><math data-latex=\"20=16\\cdot1+4\"><semantics><mrow><mn>20<\/mn><mo>=<\/mo><mn>16<\/mn><mo>\u22c5<\/mo><mn>1<\/mn><mo>+<\/mo><mn>4<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">20=16\\cdot1+4<\/annotation><\/semantics><\/math><br><math data-latex=\"16=4\\cdot4+0\"><semantics><mrow><mn>16<\/mn><mo>=<\/mo><mn>4<\/mn><mo>\u22c5<\/mo><mn>4<\/mn><mo>+<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">16=4\\cdot4+0<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p>vastaus: Suurin yhteinen tekij\u00e4 on 4<\/p>\n\n\n\n<p>c) syt(721, 679)<br><math data-latex=\"721=679\\cdot 1+42\"><semantics><mrow><mn>721<\/mn><mo>=<\/mo><mn>679<\/mn><mo>\u22c5<\/mo><mn>1<\/mn><mo>+<\/mo><mn>42<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">721=679\\cdot 1+42<\/annotation><\/semantics><\/math><br><math data-latex=\"679=42\\cdot16+7\"><semantics><mrow><mn>679<\/mn><mo>=<\/mo><mn>42<\/mn><mo>\u22c5<\/mo><mn>16<\/mn><mo>+<\/mo><mn>7<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">679=42\\cdot16+7<\/annotation><\/semantics><\/math><br><math data-latex=\"42=7\\cdot6+0\"><semantics><mrow><mn>42<\/mn><mo>=<\/mo><mn>7<\/mn><mo>\u22c5<\/mo><mn>6<\/mn><mo>+<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">42=7\\cdot6+0<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p>vastaus: Suurin yhteinen tekij\u00e4 on 7<\/p>\n<\/div>\n<\/div>\n<\/div>\n\n\n\n<div style=\"height:43px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\" style=\"font-size:18px\"><strong>Teht\u00e4v\u00e4 4.<\/strong> Selvit\u00e4 suurin yhteinen tekij\u00e4 alkuluku hajotelmien avulla. Kirjoita n\u00e4kyviin luvun alkutekij\u00e4t ja suurin yhteinen tekij\u00e4.<\/h3>\n\n\n\n<p><strong>a)<\/strong> syt(32, 18)<br><strong>b)<\/strong> syt(721, 679)<br><strong>c)<\/strong> syt(336, 99)<\/p>\n\n\n\n<div data-wp-context=\"{ &quot;autoclose&quot;: false, &quot;accordionItems&quot;: [] }\" data-wp-interactive=\"core\/accordion\" role=\"group\" class=\"wp-block-accordion is-layout-flow wp-block-accordion-is-layout-flow\">\n<div data-wp-class--is-open=\"state.isOpen\" data-wp-context=\"{ &quot;id&quot;: &quot;accordion-item-6&quot;, &quot;openByDefault&quot;: false }\" data-wp-init=\"callbacks.initAccordionItems\" data-wp-on-window--hashchange=\"callbacks.hashChange\" class=\"wp-block-accordion-item is-layout-flow wp-block-accordion-item-is-layout-flow\">\n<h3 class=\"wp-block-accordion-heading has-background has-medium-font-size\" style=\"background-color:#d6d6ff\"><button aria-expanded=\"false\" aria-controls=\"accordion-item-6-panel\" data-wp-bind--aria-expanded=\"state.isOpen\" data-wp-on--click=\"actions.toggle\" data-wp-on--keydown=\"actions.handleKeyDown\" id=\"accordion-item-6\" type=\"button\" class=\"wp-block-accordion-heading__toggle\"><span class=\"wp-block-accordion-heading__toggle-title\">Vastaus<\/span><span class=\"wp-block-accordion-heading__toggle-icon\" aria-hidden=\"true\">+<\/span><\/button><\/h3>\n\n\n\n<div inert aria-labelledby=\"accordion-item-6\" data-wp-bind--inert=\"!state.isOpen\" id=\"accordion-item-6-panel\" role=\"region\" class=\"wp-block-accordion-panel is-layout-flow wp-block-accordion-panel-is-layout-flow\">\n<p>a)<br><math data-latex=\"32=16\\cdot2=2\\cdot2\\cdot8=2\\cdot2\\cdot2\\cdot4=2\\cdot2\\cdot2\\cdot2\\cdot2=2^5\"><semantics><mrow><mn>32<\/mn><mo>=<\/mo><mn>16<\/mn><mo>\u22c5<\/mo><mn>2<\/mn><mo>=<\/mo><mn>2<\/mn><mo>\u22c5<\/mo><mn>2<\/mn><mo>\u22c5<\/mo><mn>8<\/mn><mo>=<\/mo><mn>2<\/mn><mo>\u22c5<\/mo><mn>2<\/mn><mo>\u22c5<\/mo><mn>2<\/mn><mo>\u22c5<\/mo><mn>4<\/mn><mo>=<\/mo><mn>2<\/mn><mo>\u22c5<\/mo><mn>2<\/mn><mo>\u22c5<\/mo><mn>2<\/mn><mo>\u22c5<\/mo><mn>2<\/mn><mo>\u22c5<\/mo><mn>2<\/mn><mo>=<\/mo><msup><mn>2<\/mn><mn>5<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">32=16\\cdot2=2\\cdot2\\cdot8=2\\cdot2\\cdot2\\cdot4=2\\cdot2\\cdot2\\cdot2\\cdot2=2^5<\/annotation><\/semantics><\/math><br><math data-latex=\"18=9\\cdot2=3\\cdot3\\cdot2=2\\cdot3^2\"><semantics><mrow><mn>18<\/mn><mo>=<\/mo><mn>9<\/mn><mo>\u22c5<\/mo><mn>2<\/mn><mo>=<\/mo><mn>3<\/mn><mo>\u22c5<\/mo><mn>3<\/mn><mo>\u22c5<\/mo><mn>2<\/mn><mo>=<\/mo><mn>2<\/mn><mo>\u22c5<\/mo><msup><mn>3<\/mn><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">18=9\\cdot2=3\\cdot3\\cdot2=2\\cdot3^2<\/annotation><\/semantics><\/math><br>Suurin yhteinen tekij\u00e4 on 2<\/p>\n\n\n\n<p>b)<br><math data-latex=\"721=7\\cdot103\"><semantics><mrow><mn>721<\/mn><mo>=<\/mo><mn>7<\/mn><mo>\u22c5<\/mo><mn>103<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">721=7\\cdot103<\/annotation><\/semantics><\/math><br><math data-latex=\"679=7\\cdot 97\"><semantics><mrow><mn>679<\/mn><mo>=<\/mo><mn>7<\/mn><mo>\u22c5<\/mo><mn>97<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">679=7\\cdot 97<\/annotation><\/semantics><\/math><br>Suurin yhteinen tekij\u00e4 on 7<\/p>\n\n\n\n<p>c)<br><math data-latex=\"336=2\\cdot112=3\\cdot2\\cdot56=3\\cdot2\\cdot2\\cdot28=3\\cdot2\\cdot2\\cdot2\\cdot14=3\\cdot2\\cdot2\\cdot2\\cdot2\\cdot7=2^4\\cdot3\\cdot7\"><semantics><mrow><mn>336<\/mn><mo>=<\/mo><mn>2<\/mn><mo>\u22c5<\/mo><mn>112<\/mn><mo>=<\/mo><mn>3<\/mn><mo>\u22c5<\/mo><mn>2<\/mn><mo>\u22c5<\/mo><mn>56<\/mn><mo>=<\/mo><mn>3<\/mn><mo>\u22c5<\/mo><mn>2<\/mn><mo>\u22c5<\/mo><mn>2<\/mn><mo>\u22c5<\/mo><mn>28<\/mn><mo>=<\/mo><mn>3<\/mn><mo>\u22c5<\/mo><mn>2<\/mn><mo>\u22c5<\/mo><mn>2<\/mn><mo>\u22c5<\/mo><mn>2<\/mn><mo>\u22c5<\/mo><mn>14<\/mn><mo>=<\/mo><mn>3<\/mn><mo>\u22c5<\/mo><mn>2<\/mn><mo>\u22c5<\/mo><mn>2<\/mn><mo>\u22c5<\/mo><mn>2<\/mn><mo>\u22c5<\/mo><mn>2<\/mn><mo>\u22c5<\/mo><mn>7<\/mn><mo>=<\/mo><msup><mn>2<\/mn><mn>4<\/mn><\/msup><mo>\u22c5<\/mo><mn>3<\/mn><mo>\u22c5<\/mo><mn>7<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">336=2\\cdot112=3\\cdot2\\cdot56=3\\cdot2\\cdot2\\cdot28=3\\cdot2\\cdot2\\cdot2\\cdot14=3\\cdot2\\cdot2\\cdot2\\cdot2\\cdot7=2^4\\cdot3\\cdot7<\/annotation><\/semantics><\/math><br><math data-latex=\"99=3\\cdot33=3\\cdot3\\cdot11=3^2\\cdot11\"><semantics><mrow><mn>99<\/mn><mo>=<\/mo><mn>3<\/mn><mo>\u22c5<\/mo><mn>33<\/mn><mo>=<\/mo><mn>3<\/mn><mo>\u22c5<\/mo><mn>3<\/mn><mo>\u22c5<\/mo><mn>11<\/mn><mo>=<\/mo><msup><mn>3<\/mn><mn>2<\/mn><\/msup><mo>\u22c5<\/mo><mn>11<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">99=3\\cdot33=3\\cdot3\\cdot11=3^2\\cdot11<\/annotation><\/semantics><\/math><br>Suurin yhteinen tekij\u00e4 on 3<\/p>\n<\/div>\n<\/div>\n<\/div>\n\n\n\n<div style=\"height:43px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\" style=\"font-size:18px\"><strong>Teht\u00e4v\u00e4 5.<\/strong> Selvit\u00e4 pit\u00e4v\u00e4tk\u00f6 seuraavat laskut paikkansa, perustele vastaus Eukleideen algoritmin avulla.<\/h3>\n\n\n\n<p><strong>a)<\/strong> syt(40, 24) = 4<br><strong>b)<\/strong> syt(21, 14) =7<br><strong>c)<\/strong> syt(391,&nbsp;299)=23<\/p>\n\n\n\n<div data-wp-context=\"{ &quot;autoclose&quot;: false, &quot;accordionItems&quot;: [] }\" data-wp-interactive=\"core\/accordion\" role=\"group\" class=\"wp-block-accordion is-layout-flow wp-block-accordion-is-layout-flow\">\n<div data-wp-class--is-open=\"state.isOpen\" data-wp-context=\"{ &quot;id&quot;: &quot;accordion-item-7&quot;, &quot;openByDefault&quot;: false }\" data-wp-init=\"callbacks.initAccordionItems\" data-wp-on-window--hashchange=\"callbacks.hashChange\" class=\"wp-block-accordion-item is-layout-flow wp-block-accordion-item-is-layout-flow\">\n<h3 class=\"wp-block-accordion-heading has-background\" style=\"background-color:#d6d6ff;font-size:20px\"><button aria-expanded=\"false\" aria-controls=\"accordion-item-7-panel\" data-wp-bind--aria-expanded=\"state.isOpen\" data-wp-on--click=\"actions.toggle\" data-wp-on--keydown=\"actions.handleKeyDown\" id=\"accordion-item-7\" type=\"button\" class=\"wp-block-accordion-heading__toggle\"><span class=\"wp-block-accordion-heading__toggle-title\">Vastaus<\/span><span class=\"wp-block-accordion-heading__toggle-icon\" aria-hidden=\"true\">+<\/span><\/button><\/h3>\n\n\n\n<div inert aria-labelledby=\"accordion-item-7\" data-wp-bind--inert=\"!state.isOpen\" id=\"accordion-item-7-panel\" role=\"region\" class=\"wp-block-accordion-panel is-layout-flow wp-block-accordion-panel-is-layout-flow\">\n<p>a) syt(40,24)<br><math data-latex=\"40=24\\cdot1+16\"><semantics><mrow><mn>40<\/mn><mo>=<\/mo><mn>24<\/mn><mo>\u22c5<\/mo><mn>1<\/mn><mo>+<\/mo><mn>16<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">40=24\\cdot1+16<\/annotation><\/semantics><\/math><br><math data-latex=\"24=16\\cdot1+8\"><semantics><mrow><mn>24<\/mn><mo>=<\/mo><mn>16<\/mn><mo>\u22c5<\/mo><mn>1<\/mn><mo>+<\/mo><mn>8<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">24=16\\cdot1+8<\/annotation><\/semantics><\/math><br><math data-latex=\"16=8\\cdot 2+0\"><semantics><mrow><mn>16<\/mn><mo>=<\/mo><mn>8<\/mn><mo>\u22c5<\/mo><mn>2<\/mn><mo>+<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">16=8\\cdot 2+0<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p>vastaus: <math data-latex=\"syt(40{,}24)\\ne4\"><semantics><mrow><mi>s<\/mi><mi>y<\/mi><mi>t<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>40,24<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>\u2260<\/mo><mn>4<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">syt(40{,}24)\\ne4<\/annotation><\/semantics><\/math> eli vastaus ei ole totta, koska <math data-latex=\"syt(40{,}24)=8\"><semantics><mrow><mi>s<\/mi><mi>y<\/mi><mi>t<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>40,24<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>8<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">syt(40{,}24)=8<\/annotation><\/semantics><\/math> eli lukujen suurin yhteinen tekij\u00e4 on 8.<\/p>\n\n\n\n<p>b) syt(21,14)=7<br><math data-latex=\"21=14\\cdot1+7\"><semantics><mrow><mn>21<\/mn><mo>=<\/mo><mn>14<\/mn><mo>\u22c5<\/mo><mn>1<\/mn><mo>+<\/mo><mn>7<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">21=14\\cdot1+7<\/annotation><\/semantics><\/math><br><math data-latex=\"14=7\\cdot2+0\"><semantics><mrow><mn>14<\/mn><mo>=<\/mo><mn>7<\/mn><mo>\u22c5<\/mo><mn>2<\/mn><mo>+<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">14=7\\cdot2+0<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p>vastaus: syt(21, 14) =7 on  totta, koska lukujen suurin yhteinen tekij\u00e4 on 7<\/p>\n\n\n\n<p>c) syt(391,&nbsp;299)=23<br><math data-latex=\"391=299\\cdot1+92\"><semantics><mrow><mn>391<\/mn><mo>=<\/mo><mn>299<\/mn><mo>\u22c5<\/mo><mn>1<\/mn><mo>+<\/mo><mn>92<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">391=299\\cdot1+92<\/annotation><\/semantics><\/math><br><math data-latex=\"299=92\\cdot3+23\"><semantics><mrow><mn>299<\/mn><mo>=<\/mo><mn>92<\/mn><mo>\u22c5<\/mo><mn>3<\/mn><mo>+<\/mo><mn>23<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">299=92\\cdot3+23<\/annotation><\/semantics><\/math><br><math data-latex=\"92=23\\cdot4+0\"><semantics><mrow><mn>92<\/mn><mo>=<\/mo><mn>23<\/mn><mo>\u22c5<\/mo><mn>4<\/mn><mo>+<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">92=23\\cdot4+0<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p>vastaus: syt(391,&nbsp;299)=23 on totta, koska lukujen suurin yhteinen tekij\u00e4 on 23<\/p>\n<\/div>\n<\/div>\n<\/div>\n\n\n\n<div style=\"height:43px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\" style=\"font-size:18px\"><strong>Teht\u00e4v\u00e4 6.<\/strong> Ratkaise suurin yhteinen tekij\u00e4 Eukleideen algoritmin avulla, kun lauseke on muotoa syt(syt(a, b), c).<\/h3>\n\n\n\n<p><strong>a)<\/strong> syt(syt(102, 170), 85)<br><strong>b) <\/strong>syt(syt(1144, 650), 422)<\/p>\n\n\n\n<div data-wp-context=\"{ &quot;autoclose&quot;: false, &quot;accordionItems&quot;: [] }\" data-wp-interactive=\"core\/accordion\" role=\"group\" class=\"wp-block-accordion is-layout-flow wp-block-accordion-is-layout-flow\">\n<div data-wp-class--is-open=\"state.isOpen\" data-wp-context=\"{ &quot;id&quot;: &quot;accordion-item-8&quot;, &quot;openByDefault&quot;: false }\" data-wp-init=\"callbacks.initAccordionItems\" data-wp-on-window--hashchange=\"callbacks.hashChange\" class=\"wp-block-accordion-item is-layout-flow wp-block-accordion-item-is-layout-flow\">\n<h3 class=\"wp-block-accordion-heading has-background has-medium-font-size\" style=\"background-color:#fcd8c8\"><button aria-expanded=\"false\" aria-controls=\"accordion-item-8-panel\" data-wp-bind--aria-expanded=\"state.isOpen\" data-wp-on--click=\"actions.toggle\" data-wp-on--keydown=\"actions.handleKeyDown\" id=\"accordion-item-8\" type=\"button\" class=\"wp-block-accordion-heading__toggle\"><span class=\"wp-block-accordion-heading__toggle-title\">Vastaus<\/span><span class=\"wp-block-accordion-heading__toggle-icon\" aria-hidden=\"true\">+<\/span><\/button><\/h3>\n\n\n\n<div inert aria-labelledby=\"accordion-item-8\" data-wp-bind--inert=\"!state.isOpen\" id=\"accordion-item-8-panel\" role=\"region\" class=\"wp-block-accordion-panel is-layout-flow wp-block-accordion-panel-is-layout-flow\">\n<p>a) syt(syt(102, 170), 85)<br><math data-latex=\"syt\\left(102{,}170\\right)\"><semantics><mrow><mi>s<\/mi><mi>y<\/mi><mi>t<\/mi><mrow><mo fence=\"true\" form=\"prefix\">(<\/mo><mn>102,170<\/mn><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">syt\\left(102{,}170\\right)<\/annotation><\/semantics><\/math><br><math data-latex=\"170=102\\cdot 1+68\"><semantics><mrow><mn>170<\/mn><mo>=<\/mo><mn>102<\/mn><mo>\u22c5<\/mo><mn>1<\/mn><mo>+<\/mo><mn>68<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">170=102\\cdot 1+68<\/annotation><\/semantics><\/math><br><math data-latex=\"102=68\\cdot1+34\"><semantics><mrow><mn>102<\/mn><mo>=<\/mo><mn>68<\/mn><mo>\u22c5<\/mo><mn>1<\/mn><mo>+<\/mo><mn>34<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">102=68\\cdot1+34<\/annotation><\/semantics><\/math><br><math data-latex=\"68=34\\cdot2+0\"><semantics><mrow><mn>68<\/mn><mo>=<\/mo><mn>34<\/mn><mo>\u22c5<\/mo><mn>2<\/mn><mo>+<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">68=34\\cdot2+0<\/annotation><\/semantics><\/math><br><br><math data-latex=\"syt\\left(34{,}85\\right)\"><semantics><mrow><mi>s<\/mi><mi>y<\/mi><mi>t<\/mi><mrow><mo fence=\"true\" form=\"prefix\">(<\/mo><mn>34,85<\/mn><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">syt\\left(34{,}85\\right)<\/annotation><\/semantics><\/math><br><math data-latex=\"85=34\\cdot 2+17\"><semantics><mrow><mn>85<\/mn><mo>=<\/mo><mn>34<\/mn><mo>\u22c5<\/mo><mn>2<\/mn><mo>+<\/mo><mn>17<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">85=34\\cdot 2+17<\/annotation><\/semantics><\/math><br><math data-latex=\"34=17\\cdot2+0\"><semantics><mrow><mn>34<\/mn><mo>=<\/mo><mn>17<\/mn><mo>\u22c5<\/mo><mn>2<\/mn><mo>+<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">34=17\\cdot2+0<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p>vastaus: Suurin yhteinen tekij\u00e4 on 17<\/p>\n\n\n\n<p>b) syt(syt(1144, 650), 422)<br><math data-latex=\"syt(1144,650)\"><semantics><mrow><mi>s<\/mi><mi>y<\/mi><mi>t<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>1144,650<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">syt(1144,650)<\/annotation><\/semantics><\/math><br><math data-latex=\"1144=650\\cdot1+494\"><semantics><mrow><mn>1144<\/mn><mo>=<\/mo><mn>650<\/mn><mo>\u22c5<\/mo><mn>1<\/mn><mo>+<\/mo><mn>494<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">1144=650\\cdot1+494<\/annotation><\/semantics><\/math><br><math data-latex=\"650=494\\cdot1+156\"><semantics><mrow><mn>650<\/mn><mo>=<\/mo><mn>494<\/mn><mo>\u22c5<\/mo><mn>1<\/mn><mo>+<\/mo><mn>156<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">650=494\\cdot1+156<\/annotation><\/semantics><\/math><br><math data-latex=\"494=156\\cdot3+26\"><semantics><mrow><mn>494<\/mn><mo>=<\/mo><mn>156<\/mn><mo>\u22c5<\/mo><mn>3<\/mn><mo>+<\/mo><mn>26<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">494=156\\cdot3+26<\/annotation><\/semantics><\/math><br><math data-latex=\"156=26\\cdot 6+0\"><semantics><mrow><mn>156<\/mn><mo>=<\/mo><mn>26<\/mn><mo>\u22c5<\/mo><mn>6<\/mn><mo>+<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">156=26\\cdot 6+0<\/annotation><\/semantics><\/math><br><br><math data-latex=\"syt\\left(26{,}\\ 442\\right)\"><semantics><mrow><mi>s<\/mi><mi>y<\/mi><mi>t<\/mi><mrow><mo fence=\"true\" form=\"prefix\">(<\/mo><mn>26,<\/mn><mtext>&nbsp;<\/mtext><mn>442<\/mn><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">syt\\left(26{,}\\ 442\\right)<\/annotation><\/semantics><\/math><br><math data-latex=\"422=26\\cdot16+6\"><semantics><mrow><mn>422<\/mn><mo>=<\/mo><mn>26<\/mn><mo>\u22c5<\/mo><mn>16<\/mn><mo>+<\/mo><mn>6<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">422=26\\cdot16+6<\/annotation><\/semantics><\/math><br><math data-latex=\"26=6\\cdot4+2\"><semantics><mrow><mn>26<\/mn><mo>=<\/mo><mn>6<\/mn><mo>\u22c5<\/mo><mn>4<\/mn><mo>+<\/mo><mn>2<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">26=6\\cdot4+2<\/annotation><\/semantics><\/math><br><math data-latex=\"6=2\\cdot 3+0\"><semantics><mrow><mn>6<\/mn><mo>=<\/mo><mn>2<\/mn><mo>\u22c5<\/mo><mn>3<\/mn><mo>+<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">6=2\\cdot 3+0<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p>vastaus: Suurin yhteinen tekij\u00e4 on 2<\/p>\n<\/div>\n<\/div>\n<\/div>\n\n\n\n<div style=\"height:43px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\" style=\"font-size:18px\"><strong>Teht\u00e4v\u00e4 7.<\/strong> Ratkaise x ja y Eukleideen algoritmin avulla, kun lauseke on <math data-latex=\"33x+57y=3\"><semantics><mrow><mn>33<\/mn><mi>x<\/mi><mo>+<\/mo><mn>57<\/mn><mi>y<\/mi><mo>=<\/mo><mn>3<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">33x+57y=3<\/annotation><\/semantics><\/math>.<\/h3>\n\n\n\n<div class=\"wp-block-group is-layout-constrained wp-block-group-is-layout-constrained\">\n<div data-wp-context=\"{ &quot;autoclose&quot;: false, &quot;accordionItems&quot;: [] }\" data-wp-interactive=\"core\/accordion\" role=\"group\" class=\"wp-block-accordion is-layout-flow wp-block-accordion-is-layout-flow\">\n<div data-wp-class--is-open=\"state.isOpen\" data-wp-context=\"{ &quot;id&quot;: &quot;accordion-item-9&quot;, &quot;openByDefault&quot;: false }\" data-wp-init=\"callbacks.initAccordionItems\" data-wp-on-window--hashchange=\"callbacks.hashChange\" class=\"wp-block-accordion-item is-layout-flow wp-block-accordion-item-is-layout-flow\">\n<h3 class=\"wp-block-accordion-heading has-background has-medium-font-size\" style=\"background-color:#fcd8c8\"><button aria-expanded=\"false\" aria-controls=\"accordion-item-9-panel\" data-wp-bind--aria-expanded=\"state.isOpen\" data-wp-on--click=\"actions.toggle\" data-wp-on--keydown=\"actions.handleKeyDown\" id=\"accordion-item-9\" type=\"button\" class=\"wp-block-accordion-heading__toggle\"><span class=\"wp-block-accordion-heading__toggle-title\">VINKKI!<\/span><span class=\"wp-block-accordion-heading__toggle-icon\" aria-hidden=\"true\">+<\/span><\/button><\/h3>\n\n\n\n<div inert aria-labelledby=\"accordion-item-9\" data-wp-bind--inert=\"!state.isOpen\" id=\"accordion-item-9-panel\" role=\"region\" class=\"wp-block-accordion-panel is-layout-flow wp-block-accordion-panel-is-layout-flow\">\n<p>Tee Eukleideen algoritmi \u2192 l\u00f6yd\u00e4 syt.<\/p>\n\n\n\n<p>Kirjoita joka rivi jakoj\u00e4\u00e4nn\u00f6syht\u00e4l\u00f6n\u00e4.<\/p>\n\n\n\n<p>Aloita yht\u00e4l\u00f6st\u00e4, jossa vasemmalla on syt.<\/p>\n\n\n\n<p>Korvaa esiintyv\u00e4t jakoj\u00e4\u00e4nn\u00f6kset niiden omilla yht\u00e4l\u00f6ill\u00e4.<\/p>\n\n\n\n<p>Jaa sulut, yhdist\u00e4 termit.<\/p>\n\n\n\n<p>Jatka kunnes kaikki on muodossa <math><semantics><mrow><mi>a<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi><mi>y<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">ax + by<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n<p>Lue x ja y.<\/p>\n<\/div>\n<\/div>\n<\/div>\n\n\n\n<div data-wp-context=\"{ &quot;autoclose&quot;: false, &quot;accordionItems&quot;: [] }\" data-wp-interactive=\"core\/accordion\" role=\"group\" class=\"wp-block-accordion is-layout-flow wp-block-accordion-is-layout-flow\">\n<div data-wp-class--is-open=\"state.isOpen\" data-wp-context=\"{ &quot;id&quot;: &quot;accordion-item-10&quot;, &quot;openByDefault&quot;: false }\" data-wp-init=\"callbacks.initAccordionItems\" data-wp-on-window--hashchange=\"callbacks.hashChange\" class=\"wp-block-accordion-item is-layout-flow wp-block-accordion-item-is-layout-flow\">\n<h3 class=\"wp-block-accordion-heading has-background has-medium-font-size\" style=\"background-color:#fcd8c8\"><button aria-expanded=\"false\" aria-controls=\"accordion-item-10-panel\" data-wp-bind--aria-expanded=\"state.isOpen\" data-wp-on--click=\"actions.toggle\" data-wp-on--keydown=\"actions.handleKeyDown\" id=\"accordion-item-10\" type=\"button\" class=\"wp-block-accordion-heading__toggle\"><span class=\"wp-block-accordion-heading__toggle-title\">Vastaus<\/span><span class=\"wp-block-accordion-heading__toggle-icon\" aria-hidden=\"true\">+<\/span><\/button><\/h3>\n\n\n\n<div inert aria-labelledby=\"accordion-item-10\" data-wp-bind--inert=\"!state.isOpen\" id=\"accordion-item-10-panel\" role=\"region\" class=\"wp-block-accordion-panel is-layout-flow wp-block-accordion-panel-is-layout-flow\">\n<p>33x+57y=3&nbsp;<br><math data-latex=\"syt(33, 57)\"><semantics><mrow><mi>s<\/mi><mi>y<\/mi><mi>t<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>33<\/mn><mo separator=\"true\">,<\/mo><mn>57<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">syt(33, 57)<\/annotation><\/semantics><\/math><br><math data-latex=\"57=33\\cdot1+24\"><semantics><mrow><mn>57<\/mn><mo>=<\/mo><mn>33<\/mn><mo>\u22c5<\/mo><mn>1<\/mn><mo>+<\/mo><mn>24<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">57=33\\cdot1+24<\/annotation><\/semantics><\/math><br><math data-latex=\"33=24\\cdot1+9\"><semantics><mrow><mn>33<\/mn><mo>=<\/mo><mn>24<\/mn><mo>\u22c5<\/mo><mn>1<\/mn><mo>+<\/mo><mn>9<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">33=24\\cdot1+9<\/annotation><\/semantics><\/math><br><math data-latex=\"24=9\\cdot2+6\"><semantics><mrow><mn>24<\/mn><mo>=<\/mo><mn>9<\/mn><mo>\u22c5<\/mo><mn>2<\/mn><mo>+<\/mo><mn>6<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">24=9\\cdot2+6<\/annotation><\/semantics><\/math><br><math data-latex=\"9=6\\cdot1+3\"><semantics><mrow><mn>9<\/mn><mo>=<\/mo><mn>6<\/mn><mo>\u22c5<\/mo><mn>1<\/mn><mo>+<\/mo><mn>3<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">9=6\\cdot1+3<\/annotation><\/semantics><\/math><br><math data-latex=\"6=3\\cdot2+0\"><semantics><mrow><mn>6<\/mn><mo>=<\/mo><mn>3<\/mn><mo>\u22c5<\/mo><mn>2<\/mn><mo>+<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">6=3\\cdot2+0<\/annotation><\/semantics><\/math><br><math data-latex=\"syt(33,57)=3\"><semantics><mrow><mi>s<\/mi><mi>y<\/mi><mi>t<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>33,57<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>3<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">syt(33,57)=3<\/annotation><\/semantics><\/math><br><br>jakoj\u00e4\u00e4nn\u00f6s yht\u00e4l\u00f6t:<br><math data-latex=\"24=57-33\\cdot1\"><semantics><mrow><mn>24<\/mn><mo>=<\/mo><mn>57<\/mn><mo>\u2212<\/mo><mn>33<\/mn><mo>\u22c5<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">24=57-33\\cdot1<\/annotation><\/semantics><\/math><br><math data-latex=\"9=33-24\\cdot1\"><semantics><mrow><mn>9<\/mn><mo>=<\/mo><mn>33<\/mn><mo>\u2212<\/mo><mn>24<\/mn><mo>\u22c5<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">9=33-24\\cdot1<\/annotation><\/semantics><\/math><br><math data-latex=\"6=24-9\\cdot2\"><semantics><mrow><mn>6<\/mn><mo>=<\/mo><mn>24<\/mn><mo>\u2212<\/mo><mn>9<\/mn><mo>\u22c5<\/mo><mn>2<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">6=24-9\\cdot2<\/annotation><\/semantics><\/math><br><math data-latex=\"3=9-6\\cdot1\"><semantics><mrow><mn>3<\/mn><mo>=<\/mo><mn>9<\/mn><mo>\u2212<\/mo><mn>6<\/mn><mo>\u22c5<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">3=9-6\\cdot1<\/annotation><\/semantics><\/math><br><br>Sijoitetaan yht\u00e4l\u00f6\u00f6n jakoj\u00e4\u00e4nn\u00f6syht\u00e4l\u00f6iden arvot alhaalta yl\u00f6sp\u00e4in:<br><math data-latex=\"3=9-6\\cdot1\"><semantics><mrow><mn>3<\/mn><mo>=<\/mo><mn>9<\/mn><mo>\u2212<\/mo><mn>6<\/mn><mo>\u22c5<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">3=9-6\\cdot1<\/annotation><\/semantics><\/math> <br>korvataan 6:<br><math data-latex=\"3=9-(24-9\\cdot2)\"><semantics><mrow><mn>3<\/mn><mo>=<\/mo><mn>9<\/mn><mo>\u2212<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>24<\/mn><mo>\u2212<\/mo><mn>9<\/mn><mo>\u22c5<\/mo><mn>2<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">3=9-(24-9\\cdot2)<\/annotation><\/semantics><\/math><br><math data-latex=\"3=9-24+9\\cdot2\"><semantics><mrow><mn>3<\/mn><mo>=<\/mo><mn>9<\/mn><mo>\u2212<\/mo><mn>24<\/mn><mo>+<\/mo><mn>9<\/mn><mo>\u22c5<\/mo><mn>2<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">3=9-24+9\\cdot2<\/annotation><\/semantics><\/math><br><math data-latex=\"3=3\\cdot9-24\"><semantics><mrow><mn>3<\/mn><mo>=<\/mo><mn>3<\/mn><mo>\u22c5<\/mo><mn>9<\/mn><mo>\u2212<\/mo><mn>24<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">3=3\\cdot9-24<\/annotation><\/semantics><\/math><br>korvataan 9:<br><math data-latex=\"3=3\\cdot(33-24)-24\"><semantics><mrow><mn>3<\/mn><mo>=<\/mo><mn>3<\/mn><mo>\u22c5<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>33<\/mn><mo>\u2212<\/mo><mn>24<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mn>24<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">3=3\\cdot(33-24)-24<\/annotation><\/semantics><\/math><br><math data-latex=\"3=3\\cdot33+3\\cdot(-24)-24\"><semantics><mrow><mn>3<\/mn><mo>=<\/mo><mn>3<\/mn><mo>\u22c5<\/mo><mn>33<\/mn><mo>+<\/mo><mn>3<\/mn><mo>\u22c5<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>24<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mn>24<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">3=3\\cdot33+3\\cdot(-24)-24<\/annotation><\/semantics><\/math><br><math data-latex=\"3=3\\cdot33-4\\cdot24\"><semantics><mrow><mn>3<\/mn><mo>=<\/mo><mn>3<\/mn><mo>\u22c5<\/mo><mn>33<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mo>\u22c5<\/mo><mn>24<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">3=3\\cdot33-4\\cdot24<\/annotation><\/semantics><\/math><br>korvataan 24:<br><math data-latex=\"3=3\\cdot33-4\\cdot(57-33)\"><semantics><mrow><mn>3<\/mn><mo>=<\/mo><mn>3<\/mn><mo>\u22c5<\/mo><mn>33<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mo>\u22c5<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>57<\/mn><mo>\u2212<\/mo><mn>33<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">3=3\\cdot33-4\\cdot(57-33)<\/annotation><\/semantics><\/math><br><math data-latex=\"3=3\\cdot33-4\\cdot57-4\\cdot(-33)\"><semantics><mrow><mn>3<\/mn><mo>=<\/mo><mn>3<\/mn><mo>\u22c5<\/mo><mn>33<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mo>\u22c5<\/mo><mn>57<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mo>\u22c5<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>33<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">3=3\\cdot33-4\\cdot57-4\\cdot(-33)<\/annotation><\/semantics><\/math><br><math data-latex=\"3=3\\cdot33-4\\cdot57+4\\cdot33\"><semantics><mrow><mn>3<\/mn><mo>=<\/mo><mn>3<\/mn><mo>\u22c5<\/mo><mn>33<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mo>\u22c5<\/mo><mn>57<\/mn><mo>+<\/mo><mn>4<\/mn><mo>\u22c5<\/mo><mn>33<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">3=3\\cdot33-4\\cdot57+4\\cdot33<\/annotation><\/semantics><\/math><br><math data-latex=\"3=7\\cdot33-4\\cdot57\"><semantics><mrow><mn>3<\/mn><mo>=<\/mo><mn>7<\/mn><mo>\u22c5<\/mo><mn>33<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mo>\u22c5<\/mo><mn>57<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">3=7\\cdot33-4\\cdot57<\/annotation><\/semantics><\/math><br><br>vastaus: x=7 ja y=-4<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n\n\n\n<div style=\"height:43px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\" style=\"font-size:18px\"><strong>Teht\u00e4v\u00e4<\/strong> <strong>8. <\/strong>Toteuta ohjelmallisesti Eukleideen algoritmi niin, ett\u00e4 koodi tulostaa n\u00e4kyviin jakoyht\u00e4l\u00f6n kaikki v\u00e4livaiheet sek\u00e4 suurimman yhteisen tekij\u00e4n.<\/h3>\n\n\n\n<p>Esimerkki tuloste:<\/p>\n\n\n\n<p>1819 = 867 * 2 + 85<br>867 = 85 * 10 + 17<br>85 = 17 * 5 + 0<br>Suurin yhteinen tekij\u00e4 on 17 eli toiseksi viimeinen jakoyht\u00e4l\u00f6n jakoj\u00e4\u00e4nn\u00f6s.<\/p>\n\n\n\n<div data-wp-context=\"{ &quot;autoclose&quot;: false, &quot;accordionItems&quot;: [] }\" data-wp-interactive=\"core\/accordion\" role=\"group\" class=\"wp-block-accordion is-layout-flow wp-block-accordion-is-layout-flow\">\n<div data-wp-class--is-open=\"state.isOpen\" data-wp-context=\"{ &quot;id&quot;: &quot;accordion-item-11&quot;, &quot;openByDefault&quot;: false }\" data-wp-init=\"callbacks.initAccordionItems\" data-wp-on-window--hashchange=\"callbacks.hashChange\" class=\"wp-block-accordion-item is-layout-flow wp-block-accordion-item-is-layout-flow\">\n<h3 class=\"wp-block-accordion-heading has-background has-medium-font-size\" style=\"background-color:#fcd8c8\"><button aria-expanded=\"false\" aria-controls=\"accordion-item-11-panel\" data-wp-bind--aria-expanded=\"state.isOpen\" data-wp-on--click=\"actions.toggle\" data-wp-on--keydown=\"actions.handleKeyDown\" id=\"accordion-item-11\" type=\"button\" class=\"wp-block-accordion-heading__toggle\"><span class=\"wp-block-accordion-heading__toggle-title\">Vastaus<\/span><span class=\"wp-block-accordion-heading__toggle-icon\" aria-hidden=\"true\">+<\/span><\/button><\/h3>\n\n\n\n<div inert aria-labelledby=\"accordion-item-11\" data-wp-bind--inert=\"!state.isOpen\" id=\"accordion-item-11-panel\" role=\"region\" class=\"wp-block-accordion-panel is-layout-flow wp-block-accordion-panel-is-layout-flow\">\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"452\" src=\"https:\/\/blog.edu.turku.fi\/allu\/files\/2026\/03\/Nayttokuva-2026-03-09-081829-1024x452.png\" alt=\"\" class=\"wp-image-1452\" srcset=\"https:\/\/blog.edu.turku.fi\/allu\/files\/2026\/03\/Nayttokuva-2026-03-09-081829-1024x452.png 1024w, https:\/\/blog.edu.turku.fi\/allu\/files\/2026\/03\/Nayttokuva-2026-03-09-081829-300x133.png 300w, https:\/\/blog.edu.turku.fi\/allu\/files\/2026\/03\/Nayttokuva-2026-03-09-081829-768x339.png 768w, https:\/\/blog.edu.turku.fi\/allu\/files\/2026\/03\/Nayttokuva-2026-03-09-081829.png 1032w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<p><strong>Koodi:<\/strong><\/p>\n\n\n\n<p>def toteuttaa_jakoyhtalo(a, b):<\/p>\n\n\n\n<p>&nbsp; &nbsp; r = a % b<\/p>\n\n\n\n<p>&nbsp; &nbsp; jakojaannokset = []<\/p>\n\n\n\n<p>&nbsp; &nbsp; while r != 0:<\/p>\n\n\n\n<p>&nbsp; &nbsp; &nbsp; &nbsp; q = a \/\/ b<\/p>\n\n\n\n<p>&nbsp; &nbsp; &nbsp; &nbsp; r = a % b<\/p>\n\n\n\n<p>&nbsp; &nbsp; &nbsp; &nbsp; print(f&#8221;{a} = {b} * {q} + {r}&#8221;)<\/p>\n\n\n\n<p>&nbsp; &nbsp; &nbsp; &nbsp; a = b<\/p>\n\n\n\n<p>&nbsp; &nbsp; &nbsp; &nbsp; b = r<\/p>\n\n\n\n<p>&nbsp; &nbsp; &nbsp; &nbsp; jakojaannokset.append(r)<\/p>\n\n\n\n<p>&nbsp; &nbsp; return f&#8221;Suurin yhteinen tekij\u00e4 on {jakojaannokset[len(jakojaannokset)-2]} eli toiseksi viimeinen jakoyht\u00e4l\u00f6n jakoj\u00e4\u00e4nn\u00f6s.&#8221;<\/p>\n\n\n\n<p>print(toteuttaa_jakoyhtalo(1819, 867))<\/p>\n\n\n\n<p><\/p>\n\n\n\n<p><strong>Tuloste:<\/strong><\/p>\n\n\n\n<p>1819 = 867 * 2 + 85<br>867 = 85 * 10 + 17<br>85 = 17 * 5 + 0<br>Suurin yhteinen tekij\u00e4 on 17 eli toiseksi viimeinen jakoyht\u00e4l\u00f6n jakoj\u00e4\u00e4nn\u00f6s.<\/p>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Eukleideen algoritmi Teht\u00e4v\u00e4t on jaoteltu v\u00e4rin mukaan helppoihin, perus ja vaikeisiin teht\u00e4viin. Teht\u00e4v\u00e4 1.1. Kirjoita jakoyht\u00e4l\u00f6 seuraaville jakolaskuille. a) 216\\frac {21}{6} b) 12823\\frac{128}{23} c) 4810\\frac{48}{10} Teht\u00e4v\u00e4 1.2. Mitk\u00e4 ovat kohdan 1.1. jakoyht\u00e4l\u00f6iden kokonaisosat ja jakoj\u00e4\u00e4nn\u00f6kset? Teht\u00e4v\u00e4 2. Selvit\u00e4 suurin yhteinen<\/p>\n","protected":false},"author":3745,"featured_media":0,"parent":95,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-1339","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/blog.edu.turku.fi\/allu\/wp-json\/wp\/v2\/pages\/1339","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.edu.turku.fi\/allu\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/blog.edu.turku.fi\/allu\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/blog.edu.turku.fi\/allu\/wp-json\/wp\/v2\/users\/3745"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.edu.turku.fi\/allu\/wp-json\/wp\/v2\/comments?post=1339"}],"version-history":[{"count":26,"href":"https:\/\/blog.edu.turku.fi\/allu\/wp-json\/wp\/v2\/pages\/1339\/revisions"}],"predecessor-version":[{"id":2501,"href":"https:\/\/blog.edu.turku.fi\/allu\/wp-json\/wp\/v2\/pages\/1339\/revisions\/2501"}],"up":[{"embeddable":true,"href":"https:\/\/blog.edu.turku.fi\/allu\/wp-json\/wp\/v2\/pages\/95"}],"wp:attachment":[{"href":"https:\/\/blog.edu.turku.fi\/allu\/wp-json\/wp\/v2\/media?parent=1339"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}