# Difference between revisions of "2002 AMC 12B Problems/Problem 7"

## Problem

The product of three consecutive positive integers is $8$ times their sum. What is the sum of their squares?

$\mathrm{(A)}\ 50 \qquad\mathrm{(B)}\ 77 \qquad\mathrm{(C)}\ 110 \qquad\mathrm{(D)}\ 149 \qquad\mathrm{(E)}\ 194$

## Solution

Let the three consecutive integers be $x-1, x, x+1$; then $$(x-1)(x)(x+1) = x(x^2 - 1) = 8(x-1 + x + x+1) = 24x$$

Since $x \neq 0$, we have $x^2 = 25$, with the positive solution being $x = 5$. Then $4^2 + 5^2 + 6^2 = 77\ \mathrm{(B)}$.