Let the three numbers be n 1 , n 2 , and n 3 . Let their remainders when divided by 3 be r 1 , r 2 , and r 3 . The sum n 1 + n 2 + n 3 is divisible by 3 if and only if r 1 + r 2 + r 3 is divisible by 3. Case 1: All remainders are 0. r 1 = 0, r 2 = 0, r 3 = 0 Sum of remainders = 0 + 0 + 0 = 0, which is divisible by 3. Case 2: All remainders are 1. r 1 = 1, r 2 = 1, r 3 = 1 Sum of remainders = 1 + 1 + 1 = 3, which is divisible by 3. Case 3: All remainders are 2. r 1 = 2, r 2 = 2, r 3 = 2 Sum of remainders = 2 + 2 + 2 = 6, which is divisible by 3. Case 4: One remainder is 0, one is 1, and one is 2. r 1 = 0, r 2 = 1, r 3 = 2 (in any order). Sum of remainders = 0 + 1 + 2 = 3, which is divisible by 3. The sum of three numbers is divisible by 3 if and only if all three numbers have the same remainder when divided by 3, or if they all have different remainders when divided by 3.