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 $6^{6^6}?$

$1$

$3$

$5$

$7$

$9$

Answer: C

Concepts:modular exponentiation digits

Difficulty rating: 1510

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.

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