Question

calcule el residuo de dividir E entre 25
E=24^95 +24^94 .7+24^93 .7^2+24^92 .7^3+......+24.7^94 +7^95

Answer 1

$E=24^{95}+24^{95}\times 7+24^{95}\times 7^2+\cdots +24\times 7^{94}+7^{95}\\\\E\equiv -1+7-7^2+7^3-\cdots -7^{94}+7^{95}\mod 25\\\\7^{96}-E\equiv \dfrac{7^{97}+1}{8}\mod25\\\\E\equiv7^{96}-\dfrac{7^{97}+1}{8}\mod25\\\\E\equiv49^{48}-\dfrac{7^{97}+1}{8}\mod25\\\\E\equiv1-\dfrac{7^{97}+1}{8}\mod25\\\\\text{Tenemos que }1\equiv -24\mod 25\text{ por ende}\\\\E\equiv-24+3(7^{97}+1)\mod25\\\\E\equiv1+3(7^{97}+1)\mod25\\\\----\\7^{97}=7\times 7^{96}\equiv 7\mod 25\\----$

$E\equiv1+3(7+1)\mod25\\\\E\equiv 0\mod25$

Respuesta: 0