The conversation between human and AI assistant.
What is the remainder when 3^1000 is divided by 7?
Can you explain how you arrived at that answer?
Sure. We can use Fermat's Little Theorem to find the remainder when 3^1000 is divided by 7. According to the theorem, if p is a prime number and a is any integer not divisible by p, then a^(p-1) % p = 1. In this case, p is 7 and a is 3, so 3^6 % 7 = 1. Therefore, 3^1000 % 7 is the same as (3^6)^166 % 7, which is 1^166 % 7, which is 1. But we also know that 3^1000 % 7 is equivalent to the remainder we're looking for, plus any multiple of 7. So we need to find the smallest multiple of 7 that is greater than or equal to 1, which is 7 itself. Therefore, the remainder when 3^1000 is divided by 7 is 6.