Two positive integers x and y have a greatest common divisor (GCD) of 6 and a least common… — GRE General Test

Two positive integers $x$ and $y$ have a greatest common divisor (GCD) of $6$ and a least common multiple (LCM) of $108$ . If $x = 18$ , what is the value of $y$ ?

A
$18$
$36$ Answer
C
$54$
D
$108$
E
$648$

Answer

The value of

y

36

For any two positive integers

x

and

y

, their product equals the product of their greatest common divisor and least common multiple:

x \cdot y = \text{GCD}(x,y) \cdot \text{LCM}(x,y)

. Substituting

x = 18

\text{GCD} = 6

, and

\text{LCM} = 108

gives

18y = 6 \cdot 108 = 648

. Dividing both sides by

18

yields

y = 36

Step-by-Step Solution

State the fundamental identity relating GCD, LCM, and two positive integers.

x \times y = \text{GCD}(x,y) \times \text{LCM}(x,y)

For any two positive integers, the product of the numbers is equal to the product of their greatest common divisor and least common multiple.

Substitute the known values

x = 18

\text{GCD} = 6

, and

\text{LCM} = 108

into the equation.

18 \times y = 6 \times 108

Replace the abstract variables with the specific values provided in the question stem.

Multiply the numbers on the right side of the equation.

18 \times y = 648

Simplify the product of the GCD and LCM.

Divide both sides by

18

to isolate

y

y = \frac{648}{18} = 36

Solve for the unknown integer

y

Key Concept

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

Estimated Time:1m 0s

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