If m and n are positive two-digit integers such that m < n, \text{GCD}(m, n) = 12, and \te… — GMAT

If $m$ and $n$ are positive two-digit integers such that $m < n$ , $\text{GCD}(m, n) = 12$ , and $\text{LCM}(m, n) = 420$ , what is the value of $n - m$ ?

$24$ Answer
B
$36$
C
$72$
D
$144$
E
$408$

Answer

The value of

n - m

24

Writing

m = 12x

and

n = 12y

for coprime integers

x < y

gives

\text{LCM}(m, n) = 12xy = 420

, which simplifies to

xy = 35

. The coprime factor pairs of

35

are

(1, 35)

and

(5, 7)

. The pair

(1, 35)

yields

m = 12

and

n = 420

, but

420

is not a two-digit integer. The pair

(5, 7)

gives two-digit integers

m = 60

and

n = 84

. The difference is

84 - 60 = 24

Step-by-Step Solution

Express

m

and

n

in terms of their GCD.

Let

m = 12x

and

n = 12y

, where

x

and

y

are coprime positive integers with

x < y

Factoring out the greatest common divisor

\text{GCD}(m, n) = 12

leaves coprime factors

x

and

y

Relate GCD and LCM to determine

x \cdot y

\text{LCM}(m, n) = 12 \cdot x \cdot y = 420 \implies x \cdot y = 35

For integers written as

12x

and

12y

with

\text{GCD}(x,y)=1

, the least common multiple is

12xy

Determine the valid pair

(x, y)

that satisfies the two-digit constraint.

The coprime factor pairs of

35

with

x < y

are

(1, 35)

and

(5, 7)

. For

(1, 35)

m = 12

and

n = 420

(a three-digit number). For

(5, 7)

m = 60

and

n = 84

(both are two-digit numbers). Thus,

(x, y) = (5, 7)

The problem specifies that both

m

and

n

are two-digit positive integers.

Compute

n - m

n - m = 84 - 60 = 24

Subtracting

m = 60

from

n = 84

gives

24

Key Concept

Properties of Greatest Common Divisor (GCD) and Least Common Multiple (LCM)

Question

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