Let x and y be positive integers such that \gcd(x, y) = 14 and \text{lcm}(x, y) = 420. Whi… — GRE General Test

Let $x$ and $y$ be positive integers such that $\gcd(x, y) = 14$ and $\text{lcm}(x, y) = 420$ . Which of the following values could be the sum $x + y$ ? Select all such values.

154Answer
182Answer
C
210
238Answer
E
435

Answer

The possible values for the sum

x + y

are 154, 182, and 238.

By writing

x = 14a

and

y = 14b

with

\gcd(a, b) = 1

, the relation

\text{lcm}(x, y) = 14ab = 420

requires

ab = 30

. The positive coprime factor pairs of 30 are

(1, 30)

(2, 15)

(3, 10)

, and

(5, 6)

. Multiplying these pairs by 14 gives the possible sums 434, 238, 182, and 154. Therefore, the options equal to 154, 182, and 238 are all correct.

Step-by-Step Solution

Express

x

and

y

in terms of their greatest common divisor.

Let

x = 14a

and

y = 14b

, where

a

and

b

are positive integers such that

\gcd(a, b) = 1

Factoring out the greatest common divisor leaves coprime quotient factors

a

and

b

Relate the least common multiple to

a

and

b

\text{lcm}(x, y) = 14ab = 420 \implies ab = \frac{420}{14} = 30$.

The least common multiple of two numbers sharing a GCD of

g

is given by

g \cdot a \cdot b

Find all coprime pairs

(a, b)

with

a \le b

whose product is 30.

The prime factorization of 30 is

2 \times 3 \times 5

. The valid coprime pairs

(a, b)

are

(1, 30)

(2, 15)

(3, 10)

, and

(5, 6)

Since

\gcd(a, b) = 1

, all factors of 30 split into pairs of coprime integers.

Calculate the corresponding values of

x

y

, and their sum

x + y

for each pair.

Pair (1, 30):

x = 14, y = 420 \implies x + y = 434

.
Pair (2, 15):

x = 28, y = 210 \implies x + y = 238

.
Pair (3, 10):

x = 42, y = 140 \implies x + y = 182

.
Pair (5, 6):

x = 70, y = 84 \implies x + y = 154

Multiplying each pair by the GCD of 14 yields the original integers

x

and

y

Key Concept

Relationship between GCD, LCM, and prime factorization of quotient factors

Question

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