2025 AMC 12B Problem 9

Below is the professionally curated solution for Problem 9 of the 2025 AMC 12B, 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.

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Concepts:modular exponentiation digits

Difficulty rating: 1510

What is the tens digit of $6^{6^6}?$

$1$

$3$

$5$

$7$

$9$

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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