Two positive integers, a and b, are such that a < b. The greatest common divisor of a and … — ACT

Two positive integers, $a$ and $b$ , are such that $a < b$ . The greatest common divisor of $a$ and $b$ is $12$ , and their least common multiple is $720$ . If $a$ is a multiple of $5$ but $b$ is not a multiple of $5$ , what is the value of $a$ ?

Answer: 60

Answer

By representing

a = 12x

and

b = 12y

with

\gcd(x, y) = 1

and

x < y

, the least common multiple constraint gives

12xy = 720

, which simplifies to

xy = 60

. Since

a

is a multiple of

5

and

b

is not, the factor of

5

60

must belong to

x

. Given that

x < y

and

\gcd(x, y) = 1

, the only valid coprime factorization of

60

where the factor

5

is in

x

and

x < y

x = 5

and

y = 12

. This yields

a = 12 \times 5 = 60

Step-by-Step Solution

Express the two numbers in terms of their greatest common divisor (GCD).

a = 12x

and

b = 12y

, where

\gcd(x, y) = 1

and

x < y

Since the greatest common divisor of

a

and

b

12

, both numbers must be multiples of

12

, and their remaining parts

x

and

y

must be coprime to ensure their GCD is exactly

12

Use the least common multiple (LCM) to find the product of

x

and

y

xy = 60

The least common multiple of

12x

and

12y

when

\gcd(x, y) = 1

12xy

. Setting

12xy = 720

and dividing by

12

gives

xy = 60

Identify the constraints on

x

and

y

based on the divisibility by

5

5

must divide

x

, and

5

must not divide

y

We are given that

a = 12x

is a multiple of

5

, which means

5

must be a factor of

x

. Since

b = 12y

is not a multiple of

5

5

cannot be a factor of

y

Determine the unique pair

(x, y)

that satisfies all constraints.

x = 5

and

y = 12

The product

xy = 60

has prime factorization

2^2 \times 3 \times 5

. Since

\gcd(x, y) = 1

, the factor

5

must belong to

x

, and the other prime factors

2^2

and

3

can be distributed. To satisfy

x < y

, the only possible assignment is

x = 5

and

y = 12

(since other assignments like

x = 15, y = 4

x = 20, y = 3

violate

x < y

Calculate the value of

a

a = 60

Since

a = 12x

and

x = 5

, we find

a = 12 \times 5 = 60

Key Concept

Using prime factorizations to analyze greatest common divisors and least common multiples under algebraic and inequality constraints.

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