Given: p= remainder when 784 is divided by 342 q= remainder when 784 is divided by 344 Formula used: Modular arithmetic and properties of remainders. If a≡b(bmodm), then an≡bn(bmodm) an−bn is divisible by a−b Calculations: For p : We need to find the remainder when 784 is divided by 342 . Notice that 342=73−1(. since 73=343). We know that an−bn is divisible by a−b. Consider 73≡1(bmod342) because 73−1=343−1=342, which is divisible by 342 . Now, we have 784=(73)28. ⇒784≡128(bmod342) ⇒784≡1(bmod342) So, p=1. For q: We need to find the remainder when 784 is divided by 344 . Notice that 344=73+1(. since 73=343). We know that an+bn is divisible by a+b when n is odd. We also know that an−bn is divisible by a+b when n is even. Consider 73≡−1(bmod344) because 73+1=343+1=344, which is divisible by 344. Now, we have 784=(73)28. ⇒784≡(−1)28(bmod344) ⇒784≡1(bmod344) So, q=1. Value of (p−q) : ⇒p−q=1−1 ⇒p−q=0 ∴(p−q) is equal to 0.
