Let p be the remainder when 7^84 is divided by 342 and q be the remainder when 7^84 is divided by 344. What is (p − q) equal to?

Note 342 = 7³ − 1 = 343 − 1 and 344 = 7³ + 1 = 343 + 1. We have 7^84 = (7³)^28 = 343^28. Modulo 342: 343 ≡ 1, so 343^28 ≡ 1 (mod 342), giving p = 1. Modulo 344: 343 ≡ −1, so 343^28 ≡ (−1)^28 = 1 (mod 344), giving q = 1. Therefore p − q = 0.