For two positive integers a and b, the greatest common divisor is 15 and the least common … — GMAT

For two positive integers $a$ and $b$ , the greatest common divisor is $15$ and the least common multiple is $180$ . If $a = 45$ , what is the value of $b$ ?

A
12
B
15
60Answer
D
135
E
180

Answer

For any two positive integers

a

and

b

, the product of their greatest common divisor and their least common multiple equals the product of the two numbers, expressed as

\text{GCD}(a, b) \times \text{LCM}(a, b) = a \times b

. Plugging in the given values yields

15 \times 180 = 45 \times b

. Dividing

2700

45

results in

60

Step-by-Step Solution

Recall the fundamental relation between GCD and LCM for two positive integers.

\text{GCD}(a, b) \times \text{LCM}(a, b) = a \times b

The product of the greatest common divisor and the least common multiple of two positive integers is equal to the product of the integers themselves.

Substitute the given values into the formula.

15 \times 180 = 45 \times b

We are given

\text{GCD}(a,b) = 15

\text{LCM}(a,b) = 180

, and

a = 45

Solve for

b

b = \frac{15 \times 180}{45} = \frac{180}{3} = 60

Dividing both sides by

45

simplifies to

180 \div 3 = 60

Key Concept

Relationship between GCD, LCM, and the product of two positive integers.

Estimated Time:1m 30s

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