Two numbers whose sum equals their product

Question

Two plus two equals two times two. 1. Find every pair of numbers a and b (not necessarily whole numbers) with a + b = a × b. 2. Which of those pairs consist of whole numbers? 3. Is there a triple of positive whole numbers whose sum equals their product?

Accepted Answer

Every pair of the form (a, a/(a minus 1)) with a not equal to 1, and also (0, 0). The only pair of positive whole numbers is 2 and 2. For three positive whole numbers, 1, 2 and 3 works. Solve. From a + b = ab, rearrange to ab minus b = a, so b(a minus 1) = a, giving b = a / (a minus 1) whenever a is not 1. (If a = 1 the equation reads 1 + b = b, impossible.) Examples: a = 3 gives b = 1.5 (3 + 1.5 = 4.5 = 3 × 1.5); a = 4 gives b = 4/3; a = 5 gives b = 1.25; a = minus 1 gives b = 1/2 (minus 1 + 1/2 = minus 1/2 = minus 1 × 1/2). And a = 0 gives b = 0. Whole numbers. For b = a/(a minus 1) to be a…

Two numbers whose sum equals their product

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