Multiple choice

How many different remainders can result when the100th power of an integer is divided by 125?

  1. 1

  2. 2

  3. 5

  4. 25

  5. 125
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B Correct answer
Explanation

By Euler's totient theorem or checking powers modulo 125, the 100th power of an integer modulo 125 can only be 0 (if multiple of 5) or 1 (if coprime to 5).

AI explanation

Let the integer be n. Using the binomial theorem, if n is divisible by 5, the remainder is 0. If n is not divisible by 5, the Euler totient function gives phi(125) = 100, so n^100 is congruent to 1 modulo 125. Therefore, the only possible remainders are 0 and 1. The result is 2.

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