Track / Stage 4 / 285 of 340 #545 of 1964

Problem 545

AMC 12 late, AIME early

Number theory Difficulty 4.9 Find the answer

2. The sum of the sum and the product of two positive integers is 2005, and one of them is a perfect square. Then the difference between the larger number and the smaller number is

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Official solution

$2.1001,101$ .
Let the larger of these two positive integers be $x$ , and the smaller one be $y$ , then
$x y + x + y = 2005 \text{, }$

which means $(x+1)(y+1)=2006$ .
Notice that, the only possibilities are
$(x+1)(y+1)=2 \times 1003 = 17 \times 118 \text{, }$

Therefore, $x-y=1001$ or $x-y=101$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.