Find the remainder when (257^166 - 243^166) is divided by 500.

Sure, here is the step by step solution:

Step 1: We know that (a-b) is divisible by (a^n - b^n) for any positive integer n. So, (257 - 243) = 14 is a divisor of (257^166 - 243^166).

Step 2: We need to find the remainder when (257^166 - 243^166) is divided by 500. We can use the Chinese Remainder Theorem (CRT) to break this down into two problems: finding the remainder when divided by 125 (since 500 = 5^3 * 2^2) and when divided by 4.

Step 3: When divided by 4, the remainder is 0 because both 257^166 and 243^166 are even numbers.

Step 4: When divided by 125, we can use Euler's Theorem which states that a^phi(n) ≡ 1 (mod n) if a and n are coprime. The Euler totient function phi(125) = 100, so 257^100 ≡ 1 (mod 125) and 243^100 ≡ 1 (mod 125).

Step 5: Therefore, 257^166 ≡ 257^66 (mod 125) and 243^166 ≡ 243^66 (mod 125). We can further simplify this using Euler's Theorem to get 257^66 ≡ 1 (mod 125) and 243^66 ≡ 1 (mod 125).

Step 6: So, (257^166 - 243^166) ≡ (1 - 1) ≡ 0 (mod 125).

Step 7: Using the Chinese Remainder Theorem, we can combine the results from steps 3 and 6 to find that the remainder when (257^166 - 243^166) is divided by 500 is 0.