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Number theory Difficulty 3.9 AMC 10/12 Find the answer Japan

Determine ten's place of $11^{12^{13}}$ . ( $12^{13}$ th power of $11$ , not $13$ th power of $11^{12}$ .)

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

For positive integer $n$ ,
$11^n = (10 + 1)^n = \sum_{k=0}^{n} C_n^k 10^k = n \cdot 10^n + n \cdot 10^0 = n \cdot 10^n + 1 \pmod{100}.$
Therefore, the ten's place of $11^n$ is equal to the last digit of $n$ . The last digit of $12^n$ for $n = 1, 2, \dots$ is $2, 4, 8, 6, 2, 4, \dots$ , with period $4$ , so the last digit of $12^{13}$ is $2$ and the answer is $2$ .

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