Answer to Question #313959 in Discrete Mathematics for Bij8

Question

Value of 3^222mod11

Answer 1

Since $\gcd(3,11)=1$ and $\varphi(11)=10,$ by Euler's Theorem we get that $3^{10}=1 \mod 11.$

Therefore,

$3^{222}\mod11=(3^{10})^{22}\cdot 3^2\mod 11=1^{22}\cdot 9\mod 11=9.$