What is the remainder when 7^100 is divided by 5?

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Prepare for the AMSOC 26-003 Module A Test. Utilize flashcards and multiple choice questions with hints and explanations. Ace your exam!

Multiple Choice

What is the remainder when 7^100 is divided by 5?

Explanation:
This question tests modular arithmetic and how to find a remainder by reducing the base and using a repeating pattern of powers modulo 5. Start by reducing the base: 7 ≡ 2 (mod 5), so 7^100 ≡ 2^100 (mod 5). The powers of 2 modulo 5 repeat every 4 steps: 2, 4, 3, 1, and then the cycle repeats. Because 100 is a multiple of 4, we have 2^100 ≡ 1 (mod 5). Hence 7^100 ≡ 1 (mod 5), so the remainder when divided by 5 is 1.

This question tests modular arithmetic and how to find a remainder by reducing the base and using a repeating pattern of powers modulo 5. Start by reducing the base: 7 ≡ 2 (mod 5), so 7^100 ≡ 2^100 (mod 5). The powers of 2 modulo 5 repeat every 4 steps: 2, 4, 3, 1, and then the cycle repeats. Because 100 is a multiple of 4, we have 2^100 ≡ 1 (mod 5). Hence 7^100 ≡ 1 (mod 5), so the remainder when divided by 5 is 1.

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