2024 AMC 12B 第 14 题

Split into integers coprime to $5$ and integers divisible by $5$

Since $\varphi(125) = 100,$ Euler’s theorem gives $n^{100} \equiv 1 \pmod{125}$ when $\gcd(n,5)=1;$ a multiple of $5$ has $n^{100}$ divisible by $125$

If $n$ is coprime to $5,$ then since $\varphi(125) = 100,$ Euler’s theorem gives $n^{100} \equiv 1 \pmod{125}.$ If $n$ is a multiple of $5,$ then $n^{100}$ is divisible by $5^{100},$ hence by $125,$ leaving remainder $0.$

So the only possible remainders are $0$ and $1,$ which is $2$ distinct values.

Thus, the correct answer is B.