2025 AMC 12B Problem 9

Problem 9 of 25EasierNumber Theory

What is the tens digit of

6^{6^6}?

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Solution

Here

6^6 = 46656.

For

n \ge 2,

the last two digits of

6^n

cycle with period

5

through

36, 16, 96, 76, 56.

Since

46656 \equiv 1 \pmod 5,

6^{46656}

ends in

56,

so the tens digit is

5.

Thus, the correct answer is C.

Concepts: modular exponentiation · digits

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.