Two positive integers m and n satisfy m < n. Their greatest common divisor is 18 and their… — GMAT

Two positive integers $m$ and $n$ satisfy $m < n$ . Their greatest common divisor is $18$ and their least common multiple is $1,080$ . If $m$ is not divisible by $27$ and $n$ is not divisible by $5$ , what is the value of $n - m$ ?

Answer: 126

Answer

The value of

n - m

126

By writing

m = 18a

and

n = 18b

with

\text{GCD}(a,b) = 1

, we find

a \cdot b = 60

. Evaluating all coprime factor pairs

(a, b)

shows that only

(5, 12)

satisfies all constraints (

m = 90

is not a multiple of

27

, and

n = 216

is not a multiple of

5

). The difference is

216 - 90 = 126

Step-by-Step Solution

Relate GCD, LCM, and coprime factors.

Let

m = 18a

and

n = 18b

, where

\text{GCD}(a, b) = 1

and

a < b

. Then

a \cdot b = \frac{1080}{18} = 60

The product of two positive integers equals the product of their GCD and LCM.

Determine all coprime factor pairs of

60

The coprime pairs

(a, b)

with

a < b

such that

a \cdot b = 60

are

(1, 60)

(3, 20)

(4, 15)

, and

(5, 12)

Since

\text{GCD}(a, b) = 1

, prime factors of

60 = 2^2 \cdot 3 \cdot 5

cannot be shared between

a

and

b

Apply the given constraints to eliminate invalid pairs.

The pair

(5, 12)

is the only pair satisfying both conditions:

m = 90

is not divisible by

27

, and

n = 216

is not divisible by

5

Pairs

(1,60)

and

(4,15)

yield

n

divisible by

5

; pair

(3,20)

yields

m=54

which is divisible by

27

Compute the final difference

n - m

n - m = 216 - 90 = 126

Subtract

m

from

n

using the valid values found.

Key Concept

Properties of GCD and LCM, prime factorization, and coprime factor pairs

Estimated Time:2m 0s

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