Exercice 29 --- (id : 1902)
Activités numériques I: Exercice 29
correction
1 16n+1N\dfrac{16}{n+1}\in\N
    (n+1)  divise  16\iff (n+1) \; divise\;16
    (n+1){1,2,4,8,16}\iff (n+1)\in\left\{{1,2,4,8,16}\right\}
    n{0,1,3,7,15}\iff n\in\left\{{0,1,3,7,15}\right\}
2 12n2n+1N\dfrac{12n-2}{n+1}\in\N
    12(n+1)14n+1N\iff \dfrac{12(n+1)-14}{n+1}\in\N
    1214n+1N\iff 12-\dfrac{14}{n+1}\in\N
    14n+1N\iff \dfrac{14}{n+1}\in\N et 14n+112\dfrac{14}{n+1}\leqslant 12
    n+1  divise  14\iff n+1\;divise\;14 et 14n+112\dfrac{14}{n+1}\leqslant 12
    n+1{2,7,14}\iff n+1 \in\left\{{2,7,14}\right\}
    n{1,6,13}\iff n\in\left\{{1,6,13}\right\}
3 562=4×140+2562=4\times 140+2 et 22013=22×22011=4×220112^{2013}=2^2\times 2^{2011}=4\times 2^{2011}
donc 22013+562=4(22011+140)+22^{2013}+562=4\left({2^{2011}+140}\right)+2
4 a=82b+47a=82b+47 et a<4000  et  b>47a<4000\;et\;b> 47
    a=82b+47\iff a=82b+47 et 82b+47<4000  et  b>4782b+47<4000\;et\;b> 47
    a=82b+47\iff a=82b+47 et 47<b<48,247< b< 48,2
Alors les entiers a et b sont tels que
b=48b=48 et a=82×48+47=3983a=82\times 48+47=3983