Value of 3^222mod11
Since gcd(3,11)=1\gcd(3,11)=1gcd(3,11)=1 and φ(11)=10,\varphi(11)=10,φ(11)=10, by Euler's Theorem we get that 310=1mod 11.3^{10}=1 \mod 11.310=1mod11.
Therefore,
3222mod 11=(310)22⋅32mod 11=122⋅9mod 11=9.3^{222}\mod11=(3^{10})^{22}\cdot 3^2\mod 11=1^{22}\cdot 9\mod 11=9.3222mod11=(310)22⋅32mod11=122⋅9mod11=9.
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