2024 AMC 12B Problem 14

Below is the professionally curated solution for Problem 14 of the 2024 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2024 AMC 12B solutions, or check the answer key.

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Concepts:Euler’s Totient Function modular exponentiation

Difficulty rating: 1760

14.

How many different remainders can result when the $100$ th power of an integer is divided by $125?$

$1$

$2$

$5$

$25$

$125$

Solution:

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.

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