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Question

The product N of three positive integers is 6 times their sum, and one of the integers is the sum of the other two. Find the sum of all possible values of N.

Accepted Answer

Let the three integers be a, b, c. N = abc = 6(a + b + c) and c = a + b. Then N = ab(a + b) = 6(a + b + a + b) = 12(a + b). Since a and b are positive, ab = 12 so a, b is one of 1, 12, 2, 6, 3, 4 so a + b is one of 13, 8, 7 so N is one of 12 13 = 156, 12 8 = 96, 12 7 = 84 so the answer is 156 + 96 + 84 = 336.