A SIMPLE THEORY

Number Theory Level 2

THE REMAINDER OBTAINED WHEN 7^2011 + 5^2011 IS DIVIDED BY 50 IS

1 solution

Sujoy Roy
Dec 7, 2014

$7^{2011} \mod 50$

$=7*(7^2)^{1005} \mod 50$

$=7*(50-1)^{1005}\mod 50 = 7*(-1) \mod 50 = 43$

and $5^{a} \mod 50 = 25 \forall a \ge 2$ .

So, $(7^{2011} + 5^{2011}) \mod 50 = (43+25)\mod 50 = \boxed{18}$ .