https://peda.net/id/abc2fa02470 2019-03-15T11:21:10+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/e2bdb56a498 2019-03-18T15:44:43+02:00 <img src="https://math-demo.abitti.fi/math.svg?latex=301%5Cequiv1%5Cleft(mod%5C%2010%5Cright)" alt="301\equiv1\left(mod\ 10\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=301%5E%7B301%7D%5Cequiv1%5E%7B301%7D%3D1%5Cleft(mod%5C%2010%5Cright)" alt="301^{301}\equiv1^{301}=1\left(mod\ 10\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=199%5E%7B100%7D%5Cequiv%5Cleft(-1%5Cright)%5E%7B100%7D%3D1%5Cleft(mod%5C%2010%5Cright)" alt="199^{100}\equiv\left(-1\right)^{100}=1\left(mod\ 10\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=872%5Cequiv2%5Cleft(mod%5C%2010%5Cright)" alt="872\equiv2\left(mod\ 10\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=872%5E3%5Cequiv2%5E3%3D8%5Cleft(mod%5C%2010%5Cright)" alt="872^3\equiv2^3=8\left(mod\ 10\right)"/> 2019-03-18T15:44:00+02:00 https://peda.net/id/93489920498 2019-03-18T15:27:28+02:00 a) 2<br/> b) tarkistusmerkin määräävän ehdon tulos on 105 joka ei ole kongruentti luvun 0 (mod 10) kanssa, vaan luvun 5<br/> c) <br/> 128456789123 2019-03-18T15:27:28+02:00 https://peda.net/id/b9c6aafe497 2019-03-18T15:07:04+02:00 <img src="https://math-demo.abitti.fi/math.svg?latex=46%5Cequiv1%5Cleft(mod%5C%205%5Cright)" alt="46\equiv1\left(mod\ 5\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=89%5Cequiv-1%5Cleft(mod%5C%205%5Cright)" alt="89\equiv-1\left(mod\ 5\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=46%5E%7B78%7D%2B89%5E%7B67%7D%5Cequiv1%5E%7B78%7D%2B%5Cleft(-1%5Cright)%5E%7B79%7D%3D1-1%3D0%5Cleft(mod%5C%205%5Cright)" alt="46^{78}+89^{67}\equiv1^{78}+\left(-1\right)^{79}=1-1=0\left(mod\ 5\right)"/> 2019-03-18T15:07:04+02:00 https://peda.net/id/788de57e497 2019-03-18T14:36:36+02:00 <img src="https://math-demo.abitti.fi/math.svg?latex=koska%5C%20a%5Cequiv%20b%5Cleft(mod%5C%20n%5Cright)%5C%20ja%5C%207%5Cequiv7%5Cleft(mod%5C%20n%5Cright)%7B%2C%7D%5C%20a%2B7%5Cequiv%20b%2B7%5Cleft(mod%5C%20n%5Cright)" alt="koska\ a\equiv b\left(mod\ n\right)\ ja\ 7\equiv7\left(mod\ n\right){,}\ a+7\equiv b+7\left(mod\ n\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=koska%5C%20a%5Cequiv%20b%5Cleft(mod%5C%20n%5Cright)%5C%20ja%5C%207%5Cequiv7%5Cleft(mod%5C%20n%5Cright)%7B%2C%7D%5C%203a%5Cequiv3b%5Cleft(mod%5C%20n%5Cright)" alt="koska\ a\equiv b\left(mod\ n\right)\ ja\ 7\equiv7\left(mod\ n\right){,}\ 3a\equiv3b\left(mod\ n\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=koska%5C%20a%5Cequiv%20b%5Cleft(mod%5C%20n%5Cright)%5C%20ja%5C%202%5C%20on%5C%20positiivinen%5C%20kokonaisluku%7B%2C%7D%5C%20a%5E2%5Cequiv%20b%5E2%5Cleft(mod%5C%20n%5Cright)" alt="koska\ a\equiv b\left(mod\ n\right)\ ja\ 2\ on\ positiivinen\ kokonaisluku{,}\ a^2\equiv b^2\left(mod\ n\right)"/> 2019-03-18T14:36:36+02:00 https://peda.net/id/7929b8b0497 2019-03-18T15:29:04+02:00 <img src="https://math-demo.abitti.fi/math.svg?latex=3%5E%7B284%7D%3D3%5E%7B2%5Ccdot142%7D" alt="3^{284}=3^{2\cdot142}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=9%5Cequiv1%5Cleft(mod%5C%208%5Cright)" alt="9\equiv1\left(mod\ 8\right)"/> 2019-03-18T14:29:28+02:00 https://peda.net/id/148d9346471 2019-03-15T13:01:18+02:00 a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=302%3D100%5Ccdot3%2B2" alt="302=100\cdot3+2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=302%5Cequiv2%5C%20%5Cleft(mod%5C%203%5Cright)" alt="302\equiv2\ \left(mod\ 3\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=901%3D300%5Ccdot3%2B1" alt="901=300\cdot3+1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=901%5Cequiv1%5Cleft(mod%5C%203%5Cright)" alt="901\equiv1\left(mod\ 3\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=302%5Ccdot901%5Cequiv2%5Ccdot3%3D6%5Cleft(mod%5C%203%5Cright)" alt="302\cdot901\equiv2\cdot3=6\left(mod\ 3\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=6%3D2%5Ccdot3%5C%20" alt="6=2\cdot3\ "/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=6%5Cequiv0%5Cleft(mod%5C%203%5Cright)" alt="6\equiv0\left(mod\ 3\right)"/><br/> jakojäännös on 0<br/> b)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=4%3D1%5Ccdot3%2B1" alt="4=1\cdot3+1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=4%5Cequiv1%5Cleft(mod%5C%203%5Cright)" alt="4\equiv1\left(mod\ 3\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=4%5E%7B100%7D%5Cequiv1%5E%7B100%7D%5Cleft(mod%5C%203%5Cright)" alt="4^{100}\equiv1^{100}\left(mod\ 3\right)"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=1%5E%7B100%7D%3D1" alt="1^{100}=1"/></div> <div>jakojäännös on 1</div> <div>c)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=5%3D2%5Ccdot3-1" alt="5=2\cdot3-1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=5%5Cequiv%5Cleft(-1%5Cright)%5Cleft(mod%5C%203%5Cright)" alt="5\equiv\left(-1\right)\left(mod\ 3\right)"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(-1%5Cright)%5E%7B70%7D%3D1" alt="\left(-1\right)^{70}=1"/></div> <div>jakojäännös on 1</div> </div> 2019-03-15T12:57:09+02:00 https://peda.net/id/dc99d230470 2019-03-15T12:41:16+02:00 <div><img src="https://math-demo.abitti.fi/math.svg?latex=Jos%5C%20a%5Cequiv%20b%5Cleft(mod%5C%20n%5Cright)%5C%20ja%5C%20c%5Cequiv%20d%5Cleft(mod%5C%20n%5Cright)%5C%20ja%5C%20luku%5C%20p%5C%20on%5C%20positiivinen%5C%20kokonaisluku" alt="Jos\ a\equiv b\left(mod\ n\right)\ ja\ c\equiv d\left(mod\ n\right)\ ja\ luku\ p\ on\ positiivinen\ kokonaisluku"/></div> <div>a) <img src="https://math-demo.abitti.fi/math.svg?latex=a%2Bc%5Cequiv%20b%2Bd%5Cleft(mod%5C%20n%5Cright)" alt="a+c\equiv b+d\left(mod\ n\right)"/></div> <div> </div> <div>b) <img src="https://math-demo.abitti.fi/math.svg?latex=ac%5Cequiv%20bd%5C%20%5Cleft(mod%5C%20n%5Cright)" alt="ac\equiv bd\ \left(mod\ n\right)"/></div> <div> </div> <div>c) <img src="https://math-demo.abitti.fi/math.svg?latex=a%5Ep%5Cequiv%20b%5Ep%5Cleft(mod%5C%20n%5Cright)" alt="a^p\equiv b^p\left(mod\ n\right)"/></div> <div> </div> <div>Onko luku jaollinen luvulla 8?</div> <div>Jos ei, mikä on jakojäännös?</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=89%5Ccdot805%2B60" alt="89\cdot805+60"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=7%5E%7B1249%7D%2B25" alt="7^{1249}+25"/></div> <div>Jos luku on jaollinen luvulla 8, se on kongruentti nollan kanssa (mod 8)</div> <div> </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=89%3D8%5Ccdot11%2B1%7B%2C%7D%5C%20joten%5C%2089%5Cequiv1%5Cleft(mod%5C%208%5Cright)" alt="89=8\cdot11+1{,}\ joten\ 89\equiv1\left(mod\ 8\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=805%3D8%5Ccdot100%2B5%7B%2C%7D%5C%20joten%5C%20805%5Cequiv5%5C%20%5Cleft(mod%5C%208%5Cright)" alt="805=8\cdot100+5{,}\ joten\ 805\equiv5\ \left(mod\ 8\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=60%3D7%5Ccdot8%2B4%7B%2C%7D%5C%20joten%5C%2060%5Cequiv4%5Cleft(mod%5C%208%5Cright)" alt="60=7\cdot8+4{,}\ joten\ 60\equiv4\left(mod\ 8\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=89%5Ccdot805%2B60%5Cequiv1%5Cleft(mod%5C%208%5Cright)" alt="89\cdot805+60\equiv1\left(mod\ 8\right)"/></div> <div>luku ei ole jaollinen, jakojäännös on 1</div> <div> </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=7%3D1%5Ccdot8-1%7B%2C%7D%5C%207%5Cequiv-1%5C%20%5Cleft(mod%5C%208%5Cright)" alt="7=1\cdot8-1{,}\ 7\equiv-1\ \left(mod\ 8\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=25%3D8%5Ccdot3%2B1%7B%2C%7D%5C%2025%5Cequiv1%5C%20%5Cleft(mod%5C%208%5Cright)" alt="25=8\cdot3+1{,}\ 25\equiv1\ \left(mod\ 8\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=7%5E%7B1249%7D%2B25%5Cequiv%5Cleft(-1%5Cright)%5E%7B1249%7D%2B1%3D-1%2B1%3D0" alt="7^{1249}+25\equiv\left(-1\right)^{1249}+1=-1+1=0"/><br/> luku on jaollinen luvulla 8</div> <div> </div> <div>Kongruenssin avulla voidaan määrittää isojenkin potenssimuotoisten lukujen viimeinen numero</div> <div>Luvun viimeinen numero on jakojäännös, kun luku jaetaan luvulla 10</div> <div> </div> 2019-03-15T12:41:16+02:00