2024 AMC 10B Problem 18

Here $125 = 5^3$ and $\varphi(125) = 100.$ If $\gcd(n, 5) = 1,$ Euler’s theorem gives $n^{100} \equiv 1 \pmod{125}.$ And if $n$ is divisible by $5,$ then $n^{100}$ carries a factor of $5^{100},$ hence of $125,$ so $n^{100} \equiv 0 \pmod{125}.$ That leaves only two possible remainders, $0$ and $1.$ Therefore, the answer is B.