Question

Pick two consecutive positive 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?

$2$

$64$

$79$

$96$

$131$

Solution:

Let the consecutive positive integers be $x$ and $x+1$ . Their sum is $2x+1$ , which is less than $100$ .

The difference of their squares is $(x+1)^2-x^2=(x+1+x)(x+1-x)=2x+1.$

So the difference must be an odd number less than $100$ . Among the choices, $79$ is the only possibility, and it occurs for $39$ and $40$ .

Thus, C is the correct answer.

2007 AMC 8 Problem 19

Solution:

Problem 19 in Other Years