What is the remainder when ${16}^{ 32 }$ is divided by 11.

The answer is 3.

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Note $5^5=(25)^2\cdot 5\equiv 3^2\cdot 5=45\equiv 1\pmod{11}$ . Then

$16^{32}\equiv5^{32}=(5^5)^65^2\equiv 1^6\cdot 25\equiv \boxed{3}\pmod{11}$ .