AMC12 2025 B

AMC12 2025 B · Q9

AMC12 2025 B · Q9. It mainly tests Remainders & modular arithmetic, Powers & residues.

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

$6^{6^6}$ 的十位数字是多少？

(A) 1 1

(B) 3 3

(D) 7 7

(E) 9 9

Answer

Correct choice: (C)

正确答案：（C）

Solution

We wish to find $6^{6^6}\pmod{100}$ and find the tens digit. By the Chinese Remainder Theorem, it suffices to find $6^{6^6}\pmod{4}$ and $6^{6^6}\pmod{25}$ and find the solution to the system. We know that $6^{6^6}\equiv 0\pmod{4}$, so we just need to find the remainder modulo $25$. By Euler’s Totient Theorem, $6^{6^6}\equiv 6^r\pmod{25}$, where $r\equiv 6^6\pmod{\varphi(25)=20}$. Hence we need to find $6^6\pmod{20}$. Now we again use the Chinese Remainder Theorem. We know that $6^6\equiv 0\pmod{4}$ and $6^6\equiv 1^6\equiv 1\pmod{5}$. Testing cases yields $6^6\equiv 16\pmod{20}$. Therefore, we know that $6^{6^6}\equiv 6^{16}\pmod{25}$. Now we can simplify: \[6^{6^6}\equiv 6^{16}\equiv (6^4)^4\equiv 1296^4\equiv (-4)^4\equiv 256\equiv 6\pmod{25}\] Solving the system $6^{6^6}\equiv 6\pmod{25}$ and $6^{6^6}\equiv 0\pmod{4}$ yields $6^{6^6}\equiv 56\pmod{100}$. Hence the answer is $\boxed{\textbf{(C)}~5}$.

我们希望求 $6^{6^6}\pmod{100}$ 并找出十位数字。由中国剩余定理，求 $6^{6^6}\pmod{4}$ 和 $6^{6^6}\pmod{25}$ 即可。我们知道 $6^{6^6}\equiv 0\pmod{4}$，所以只需求模 $25$ 的剩余。由 Euler totient 定理，$6^{6^6}\equiv 6^r\pmod{25}$，其中 $r\equiv 6^6\pmod{\varphi(25)=20}$。故需求 $6^6\pmod{20}$。再次用中国剩余定理。我们知道 $6^6\equiv 0\pmod{4}$ 且 $6^6\equiv 1^6\equiv 1\pmod{5}$。测试情况得出 $6^6\equiv 16\pmod{20}$。因此，$6^{6^6}\equiv 6^{16}\pmod{25}$。现在简化： \[6^{6^6}\equiv 6^{16}\equiv (6^4)^4\equiv 1296^4\equiv (-4)^4\equiv 256\equiv 6\pmod{25}\] 解系统 $6^{6^6}\equiv 6\pmod{25}$ 和 $6^{6^6}\equiv 0\pmod{4}$ 得出 $6^{6^6}\equiv 56\pmod{100}$。故十位数字为 $\boxed{\textbf{(C)}~5}$。

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