Two positive integers x and y have a greatest common divisor (GCD) of 14 and a least commo… — GMAT

Two positive integers $x$ and $y$ have a greatest common divisor (GCD) of $14$ and a least common multiple (LCM) of $840$ . If $14 < x < y < 200$ and $x$ is not a multiple of $4$ , what is the value of $y - x$ ?

A
$42$
$98$ Answer
C
$112$
D
$126$
E
$238$

Answer

The value of

y - x

98

The prime factorizations of

\text{GCD}(x,y)=14=2^1 \times 7^1

and

\text{LCM}(x,y)=840=2^3 \times 3^1 \times 5^1 \times 7^1

require

x

and

y

to split the exponents of

2, 3, 5, 7

. Since

x

is not a multiple of

4

x

contains

2^1

and

y

contains

2^3

. Both contain

7^1

. To satisfy

14 < x < y < 200

x

must take factor

5

(

x = 70

) and

y

must take factor

3

(

y = 168

). The difference is

168 - 70 = 98

Step-by-Step Solution

Express the GCD and LCM in prime factorized form.

\text{GCD}(x, y) = 14 = 2^1 \times 3^0 \times 5^0 \times 7^1

and

\text{LCM}(x, y) = 840 = 2^3 \times 3^1 \times 5^1 \times 7^1

Prime factorization allows us to analyze the min and max prime exponents shared between

x

and

y

Determine the distribution of prime factor exponents between

x

and

y

For factor

7

: both

x

and

y

must have exponent

1

.
For factor

2

\min(a_x, a_y) = 1

and

\max(a_x, a_y) = 3

. Since

x

is not divisible by

4

(

2^2

a_x = 1

, which forces

a_y = 3

.
For factors

3

and

5

: one integer receives

3^1

and the other receives

5^1

The GCD takes the minimum exponent and the LCM takes the maximum exponent for each prime factor.

Test possible assignments of factors

3

and

5

under the given inequalities.

Case 1:

x

gets

3

and

y

gets

5

x = 2 \times 7 \times 3 = 42

y = 8 \times 7 \times 5 = 280

. This fails

y < 200

.
Case 2:

x

gets

5

and

y

gets

3

x = 2 \times 7 \times 5 = 70

y = 8 \times 7 \times 3 = 168

. This satisfies

14 < 70 < 168 < 200

Only Case 2 satisfies all structural and boundary constraints (

14 < x < y < 200

and

x

not a multiple of

4

Compute

y - x

y - x = 168 - 70 = 98

Direct subtraction of the valid integer values.

Key Concept

Prime Factorization Rules for GCD and LCM

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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