# Two Consecutive Numbers with Small Sum

This is problem 19 from AMC 8, 2007.

Pick two consecutive integers whose sum is less than 100. Square both of those integers and then find the difference of the squares. Which of the following could be the difference? 2, 64, 79, 96, 131.

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### Solution

The key to the solution is the well-known *formula for the differences of two squares*

a² - b² = (a - b)(a + b).

Let a be the larger of the two: a > b, implying

Of the two consecutive numbers one is odd and one is even, implying that the sum of the two is odd. Among the four remaining possibilities there is only one odd number, 79,2,64,79,96,131, which is necessarily is the answer to the problem.

We may actually find the two numbers, although this is not required to solve the problem. The smallest of the two, b, is found from

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