Track / Stage 5 / 99 of 400 #699 of 1964

Problem 699

AIME late

Number theory Difficulty 5.3 Find the answer

17. The greatest common divisor of two natural numbers is 12, and their least common multiple is 672. What are these numbers if it is known that the smaller of them is divisible by 7, and the larger is not?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

17. Let $a$ and $b$ be the sought numbers. Since $D(a, b) = 12$ , it follows that $a = 12x$ and $b = 12y$ , where $x$ and $y$ are relatively prime natural numbers, i.e., $D(x, y) = 1$ .

Given the equality $D(a, b) \cdot V(a, b) = a \cdot b$ , we have $12 \cdot 672 = 12x \cdot 12y$ , which simplifies to

$12xy = 672$ , i.e., $xy = 56$ .

We now have the following possibilities:

$x$	1	2	4	7
$y$	56	28	14	8
$D(x, y)$	1	2	2	1

Since the numbers

x

and

y

are relatively prime, and given that the smaller of them is divisible by 7, while the larger is not, we conclude that the only solution is obtained for

x = 7

and

y = 8

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