Multiple choice general knowledge math & puzzles

Find the remainder when 205^205 is divided by 101.

  1. 41

  2. 49

  3. 92

  4. 3

  5. 98
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A Correct answer
Explanation

Using Fermat's Little Theorem: for prime p=101, a^100 ≡ 1 (mod 101) when a is not divisible by 101. Therefore, 205^205 ≡ 205^(2×100 + 5) ≡ (205^100)^2 × 205^5 ≡ 1^2 × 205^5. Simplify: 205 ≡ 3 (mod 101), so 3^5 = 243 ≡ 41 (mod 101). The remainder is 41.

AI explanation

Since 205 = 2×101 + 3, we have 205 ≡ 3 (mod 101). By Fermat's Little Theorem, since 101 is prime, 3^100 ≡ 1 (mod 101), and 205 mod 100 = 5, so 3^205 ≡ 3^5 (mod 101). Computing 3^5 = 243, and 243 − 2×101 = 41, giving a remainder of 41.

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