Question

What is the remainder of (2^91 x 6^27) divided by 7?

Answer 1

$2\equiv2\mod7$
$2^2\equiv4\mod7$
$2^3\equiv8\equiv1\mod7$

$91=3(30)+1$
$\implies2^{91}\equiv(2^3)^{30}\times2^1\equiv1^{30}\times2\equiv2\mod7$

$6\equiv-1\mod7$
$6^2=6\times6\equiv6(-1)\equiv-6\equiv1\mod7$

$27=2(13)+1$
$\implies6^{27}\equiv(6^2)^{13}\times6^1\equiv1^{13}\times6\equiv6\mod7$

$\implies2^{91}\times6^{27}\equiv2\times6\equiv12\equiv5\mod7$