2025 AMC 12B Problem 9

Attempt Problem 9 of the 2025 AMC 12B below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2025 AMC 12B solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

What is the tens digit of

6^{6^6}?

1

3

5

7

9

The tens digit is determined by

6^{6^6} \bmod 100

For

n \ge 2,

6^n \bmod 100

cycles with period

5;

reduce the exponent

6^6

modulo

5

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.

Problem 9 in other years