https://peda.net/id/d3b1827a419 2019-03-08T13:10:48+02:00 https://peda.net/:static/537/peda.net.logo.bg.svg <div class="license">Tämän sivun lisenssi <a rel="license" href="https://peda.net/info">Peda.net-yleislisenssi</a></div> https://peda.net/id/ba2a2c864ca 2019-03-22T15:16:39+02:00 a) <br/> <img src="https://math-demo.abitti.fi/math.svg?latex=9%5Cequiv16%5Cequiv23%5Cequiv2%5C%20%5Cleft(mod%5C%207%5Cright)" alt="9\equiv16\equiv23\equiv2\ \left(mod\ 7\right)"/><br/> b)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2%5Ccdot2%5Cequiv2%5Ccdot5%5Cequiv2%5Ccdot8%5Cequiv1%5C%20%5Cleft(mod%5C%203%5Cright)" alt="2\cdot2\equiv2\cdot5\equiv2\cdot8\equiv1\ \left(mod\ 3\right)"/><br/> c)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=5%5Ccdot1%5Cequiv5%5Ccdot3%5Cequiv5%5Ccdot5%5Cequiv3%5C%20%5Cleft(mod%5C%202%5Cright)" alt="5\cdot1\equiv5\cdot3\equiv5\cdot5\equiv3\ \left(mod\ 2\right)"/> 2019-03-22T15:16:39+02:00 https://peda.net/id/6d1859504ca 2019-03-22T15:07:20+02:00 <img src="https://math-demo.abitti.fi/math.svg?latex=122%3D7%5Ccdot17%2B3" alt="122=7\cdot17+3"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=0%5Ccdot17%2B3%3D3" alt="0\cdot17+3=3"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=122%5Cequiv3%5C%20%5Cleft(mod%5C%2017%5Cright)" alt="122\equiv3\ \left(mod\ 17\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=-14%3D-2%5Ccdot9%2B4" alt="-14=-2\cdot9+4"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=0%5Ccdot9%2B4%3D4" alt="0\cdot9+4=4"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=-14%5Cequiv4%5C%20%5Cleft(mod%5C%209%5Cright)" alt="-14\equiv4\ \left(mod\ 9\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=34%5Cequiv13%5C%20%5Cleft(mod%5C%20n%5Cright)" alt="34\equiv13\ \left(mod\ n\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=n%3D3" alt="n=3"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=34%3D11%5Ccdot3%2B1" alt="34=11\cdot3+1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=13%3D4%5Ccdot3%2B1" alt="13=4\cdot3+1"/> 2019-03-22T15:07:20+02:00 https://peda.net/id/ed375e8e4ca 2019-03-22T15:03:45+02:00 a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=23%3D5%5Ccdot4%2B3" alt="23=5\cdot4+3"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=47%3D11%5Ccdot4%2B3" alt="47=11\cdot4+3"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=23%5Cequiv47%5C%20%5Cleft(mod%5C%204%5Cright)" alt="23\equiv47\ \left(mod\ 4\right)"/><br/> b)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=27%3D3%5Ccdot7%2B6" alt="27=3\cdot7+6"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=58%3D8%5Ccdot7%2B2" alt="58=8\cdot7+2"/><br/> c)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=281%3D93%5Ccdot3%2B2" alt="281=93\cdot3+2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=275%3D91%5Ccdot3%2B2" alt="275=91\cdot3+2"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=281%5Cequiv275%5C%20%5Cleft(mod%5C%203%5Cright)" alt="281\equiv275\ \left(mod\ 3\right)"/></div> 2019-03-22T15:03:45+02:00 https://peda.net/id/1c7133384ca 2019-03-22T14:58:19+02:00 <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Barray%7D%7Bl%7Cl%7D%0AM%26V%26M%5Cvee%20V%26%5Cneg%20M%5Cvee%5Cneg%20V%26M%5CRightarrow%20V%5C%5C%0A%5Chline%0A1%261%261%260%261%5C%5C%0A1%260%261%261%260%5C%5C%0A0%261%261%261%261%5C%5C%0A0%260%260%261%260%0A%5Cend%7Barray%7D" alt="\begin{array}{l|l}&#10;M&amp;V&amp;M\vee V&amp;\neg M\vee\neg V&amp;M\Rightarrow V\\&#10;\hline&#10;1&amp;1&amp;1&amp;0&amp;1\\&#10;1&amp;0&amp;1&amp;1&amp;0\\&#10;0&amp;1&amp;1&amp;1&amp;1\\&#10;0&amp;0&amp;0&amp;1&amp;0&#10;\end{array}"/><br/> <br/> Mikko on syytön</div> <div>Ville on syyllinen</div> 2019-03-22T14:57:55+02:00 https://peda.net/id/d6658c404ca 2019-03-22T14:55:58+02:00 <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Barray%7D%7Bl%7Cl%7D%0AA%26B%26A%5Cvee%5Cneg%20B%26A%5CRightarrow%20B%26B%5CRightarrow%20A%26A%5Cwedge%5Cneg%20B%5C%5C%0A%5Chline%0A1%261%261%261%261%260%5C%5C%0A1%260%261%260%261%261%5C%5C%0A0%261%260%261%260%260%5C%5C%0A0%260%261%261%261%260%0A%5Cend%7Barray%7D" alt="\begin{array}{l|l}&#10;A&amp;B&amp;A\vee\neg B&amp;A\Rightarrow B&amp;B\Rightarrow A&amp;A\wedge\neg B\\&#10;\hline&#10;1&amp;1&amp;1&amp;1&amp;1&amp;0\\&#10;1&amp;0&amp;1&amp;0&amp;1&amp;1\\&#10;0&amp;1&amp;0&amp;1&amp;0&amp;0\\&#10;0&amp;0&amp;1&amp;1&amp;1&amp;0&#10;\end{array}"/><br/> <br/> a ja c -kohtien lauseet ovat loogisesti ekvivalentteja</div> 2019-03-22T14:55:58+02:00 https://peda.net/id/a744d6144a3 2019-03-19T14:08:06+02:00 a) nelikulmio on suorakulmio, jos nelikulmion kaikki kulmat ovat yhtä suuret<br/> b) nelikulmion kaikki sivut ovat yhtä pitkät, jos nelikulmio on neliö<br/> c) nelikulmio on suorakulmio jos ja vain jos nelikulmion kaikki kulmat ovat yhtä suuret<br/> d) nelikulmio on neliö, jos sekä nelikulmion kaikki kulmat ovat yhtä suuret että nelikulmion kaikki sivut ovat yhtä pitkät 2019-03-19T14:08:06+02:00 https://peda.net/id/d4119b34471 2019-03-15T13:17:33+02:00 a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=3972%3D1324%5Ccdot3%2B0" alt="3972=1324\cdot3+0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=3576%3D1192%5Ccdot3%2B0" alt="3576=1192\cdot3+0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=564%3D188%5Ccdot3%2B0" alt="564=188\cdot3+0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=3972-3576%5Ccdot576%5Cequiv0-0%5Ccdot0%3D0%5Cleft(mod%5C%206%5Cright)" alt="3972-3576\cdot576\equiv0-0\cdot0=0\left(mod\ 6\right)"/><br/> jakojäännös 0<br/> b)<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=87%3D17%5Ccdot5%2B2" alt="87=17\cdot5+2"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=87%5Cequiv2%5Cleft(mod%5C%205%5Cright)" alt="87\equiv2\left(mod\ 5\right)"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=109%3D21%5Ccdot5%2B4" alt="109=21\cdot5+4"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=109%5Cequiv4%5C%20%5Cleft(mod%5C%205%5Cright)" alt="109\equiv4\ \left(mod\ 5\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=3%3D1%5Ccdot5-2" alt="3=1\cdot5-2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=4%3D1%5Ccdot5-1" alt="4=1\cdot5-1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=87%5Ccdot109%2B3%5Ccdot4%5E%7B23%7D%5Cequiv2%5Ccdot4-2%5Ccdot%5Cleft(-1%5Cright)%5E%7B23%7D%3D8%2B2%3D10%3D2%5Ccdot5%2B0%5C%20%5Cleft(mod%5C%205%5Cright)" alt="87\cdot109+3\cdot4^{23}\equiv2\cdot4-2\cdot\left(-1\right)^{23}=8+2=10=2\cdot5+0\ \left(mod\ 5\right)" title=""/><br/> jakojäännös 0 2019-03-15T13:16:49+02:00 https://peda.net/id/d7c7c592471 2019-03-15T13:09:46+02:00 a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=92%3D15%5Ccdot6%2B2" alt="92=15\cdot6+2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=92%5Cequiv2%5C%20%5Cleft(mod%5C%206%5Cright)" alt="92\equiv2\ \left(mod\ 6\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=903%3D150%5Ccdot6%2B3" alt="903=150\cdot6+3"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=903%5Cequiv3%5Cleft(mod%5C%206%5Cright)" alt="903\equiv3\left(mod\ 6\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=92%5Ccdot903%5Cequiv2%5Ccdot3%5Cequiv0%5Cleft(mod%5C%206%5Cright)" alt="92\cdot903\equiv2\cdot3\equiv0\left(mod\ 6\right)"/><br/> b)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=13%3D2%5Ccdot6%2B1" alt="13=2\cdot6+1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=5%3D0%5Ccdot6%2B5" alt="5=0\cdot6+5"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=13%5E%7B19%7D%2B5%5Cequiv1%5E%7B19%7D%2B5%3D1%2B5%3D0%5Cleft(mod%5C%206%5Cright)" alt="13^{19}+5\equiv1^{19}+5=1+5=0\left(mod\ 6\right)"/><br/> c)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=7%3D1%5Ccdot6%2B1" alt="7=1\cdot6+1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=-1%3D0%5Ccdot6-1" alt="-1=0\cdot6-1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=7%5E%7B12%7D-1%5Cequiv1%5E%7B12%7D-1%3D1-1%3D0%5Cleft(mod%5C%206%5Cright)" alt="7^{12}-1\equiv1^{12}-1=1-1=0\left(mod\ 6\right)"/> 2019-03-15T13:09:46+02:00 https://peda.net/id/628e7a88419 2019-03-08T14:00:35+02:00 a)<br/> <a href="https://peda.net/p/oskari.lahtinen/mljt/pt/153/sieppaa-png#top" title="Sieppaa.PNG"><img src="https://peda.net/p/oskari.lahtinen/mljt/pt/153/sieppaa-png:file/photo/ffd49f9a833d868005b456e3dd21873f7790a0ea/Sieppaa.PNG" alt="" title="Sieppaa.PNG" class="inline" loading="lazy"/></a><br/> x=1, y=7<br/> x=2, y=4<br/> b)<br/> <a href="https://peda.net/p/oskari.lahtinen/mljt/pt/153/sieppaa-png2#top" title="Sieppaa.PNG"><img src="https://peda.net/p/oskari.lahtinen/mljt/pt/153/sieppaa-png2:file/photo/85e52384e08c82d60c6abb1c09877ab692f03663/Sieppaa.PNG" alt="" title="Sieppaa.PNG" class="inline" loading="lazy"/></a><br/> x=2, y=1<br/> x=4, y=2<br/> c)<br/> <a href="https://peda.net/p/oskari.lahtinen/mljt/pt/153/sieppaa-png3#top" title="Sieppaa.PNG"><img src="https://peda.net/p/oskari.lahtinen/mljt/pt/153/sieppaa-png3:file/photo/35587a249b3bcad7bddfcfbaca869b8dac48c151/Sieppaa.PNG" alt="" title="Sieppaa.PNG" class="inline" loading="lazy"/></a><br/> x=2, y=2<br/> x=6, y=5 2019-03-08T13:57:45+02:00 https://peda.net/id/9c7e3e00419 2019-03-08T13:52:12+02:00 a)<br/> <div>syt(108,72)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=108%3D72%5Ccdot1%2B36" alt="108=72\cdot1+36"/></div> <div>syt(108,72)= syt(72,36)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=72%3D36%5Ccdot2" alt="72=36\cdot2"/></div> <div>syt(108,72)=36<br/> b)<br/> <div>syt(30,8)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=30%3D8%5Ccdot3%2B6" alt="30=8\cdot3+6"/></div> <div>syt(30,8)=syt(8,6)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=8%3D6%5Ccdot1%2B2" alt="8=6\cdot1+2"/> </div> <div>syt(30,8)=syt(8,6)=syt(6,2)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=6%3D2%5Ccdot3" alt="6=2\cdot3"/><br/> syt(30,8)=2<br/> c)<br/> <div>syt(192,126)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=192%3D126%5Ccdot1%2B66" alt="192=126\cdot1+66"/></div> <div>syt(192,126)=syt(126,66)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=126%3D66%5Ccdot1%2B60" alt="126=66\cdot1+60"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=66%3D60%5Ccdot1%2B6" alt="66=60\cdot1+6"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=60%3D6%5Ccdot10" alt="60=6\cdot10"/></div> <div>syt(192,126)=6</div> <div> </div> </div> </div> 2019-03-08T13:52:12+02:00