Two positive integers a and b are in the ratio 3 : 8. If the least common multiple (LCM) o… — GMAT

Two positive integers $a$ and $b$ are in the ratio $3 : 8$ . If the least common multiple (LCM) of $a$ and $b$ is $360$ , what is the greatest common divisor (GCD) of $a$ and $b$ ?

Answer: 15

Answer

The greatest common divisor of

a

and

b

is 15.

Any two positive integers in the ratio

3 : 8

can be written as

3g

and

8g

, where

g

is their greatest common divisor. Because

3

and

8

are coprime (their GCD is

1

), the least common multiple of

3g

and

8g

3 \times 8 \times g = 24g

. Setting

24g = 360

and dividing by

24

yields

g = 15

Step-by-Step Solution

Represent the two integers using their ratio and their greatest common divisor.

Let

g = \gcd(a, b)

. Then

a = 3g

and

b = 8g

, where

3

and

8

share no common factors other than

1

When two numbers are in reduced ratio

p : q

, dividing both by their GCD leaves coprime factors

p

and

q

Express the LCM of

a

and

b

in terms of

g

\text{LCM}(a, b) = 3 \times 8 \times g = 24g.

The LCM of two numbers

p \cdot g

and

q \cdot g

with

\gcd(p, q) = 1

p \cdot q \cdot g

Solve for

g

using the given LCM value of

360

24g = 360 \implies g = 15.

Dividing the given LCM by the product of the coprime ratio components yields the GCD.

Key Concept

Relationship between GCD, LCM, and coprime factor ratios of two positive integers

Estimated Time:1m 15s

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