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x and y are positive integers. If the greatest common divisor of 3x and 3y is 6, what is the value of y?

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by BTGModeratorVI » Fri Aug 14, 2020 1:10 pm

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x and y are positive integers. If the greatest common divisor of 3x and 3y is 6, what is the value of y?

(1) The greatest common divisor of 2x and 2y is 2y
(2) The least common multiple of 2x and 2y is 20

Answer: A
Source: GMAT Prep Now
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Source: — Data Sufficiency |
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Re: x and y are positive integers. If the greatest common divisor of 3x and 3y is 6, what is the value of y?

by deloitte247 » Sat Aug 15, 2020 1:02 pm
From the question stem, x and y are said to be positive integers in which the greatest common divisors of 3x and 3y = 6. This means x and y have a common factor which is 6/3 = 2

Target question => what is the value of y?

Statement 1 => The greatest common divisor of 2x and 2y is 2y
This means that 2x is dividable by 2y and x is a multiple of y
Since the only common factor of x and y is 2 then y = 2
Statement 1 is SUFFICIENT

Statement 2 => The least common multiple of 2x and 2y is 20
LCM of 2x and 2y = 20
LCM of x and y = 10
Therefore, both x and y are multiples of 2 so the value of x and y can be 10 and 2 or 2 and 10 because LCM of 2(2) and 2(10) and LCM of 2(10) and 2(2) are both 20
The value of y is either 2 or 10. This statement is not enough to arrive at a definite answer and cannot answer the target question. Statement 2 is NOT SUFFICIENT

Since only statement 1 is SUFFICIENT,
Answer = A
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Re: x and y are positive integers. If the greatest common divisor of 3x and 3y is 6, what is the value of y?

by Brent@GMATPrepNow » Sun Aug 16, 2020 5:31 am
BTGModeratorVI wrote: ↑
Fri Aug 14, 2020 1:10 pm
 x and y are positive integers. If the greatest common divisor of 3x and 3y is 6, what is the value of y?

(1) The greatest common divisor of 2x and 2y is 2y
(2) The least common multiple of 2x and 2y is 20

Answer: A
Source: GMAT Prep Now
Target question: What is the value of y?

Given: The greatest common divisor of 3x and 3y is 6
This means that, if we examine the prime factorization of 3x and prime factorization of 3y, they will share exactly ONE 3 and ONE 2.
That is:
3x = (2)(3)(?)(?)(?)(?)
3y = (2)(3)(?)(?)(?)(?)
NOTE: Both prime factorizations might include other primes, BUT there is no additional overlap beyond the ONE 3 and ONE 2.

Notice that if we divide both sides of both prime factorizations by 3, we get:
x = (2)(?)(?)(?)(?)
y = (2)(?)(?)(?)(?)

This tells us that the greatest common divisor (GCD) of x and y is 2.

Statement 1: The greatest common divisor of 2x and 2y is 2y
We already know that...
x = (2)(?)(?)(?)(?)
y = (2)(?)(?)(?)(?)

So,
2x = (2)(2)(?)(?)(?)(?)
2y = (2)(2)(?)(?)(?)(?)
This tells us that the greatest common divisor (GCD) of 2x and 2y =(2)(2) = 4.

The statement tells us that the GCD of 2x and 2y is 2y, which means 2y = 4
Solve the equation to get y = 2. PERFECT!!
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: The least common multiple of 2x and 2y is 20
There are several values of x and y that satisfy statement 2 (as well as satisfying the given information). Here are two:
Case a: x = 2 and y = 10. In this case, 2x = 4 and 2y = 20, and the least common multiple of 4 and 20 is 20. Also notice that 3x = 6 and 3y = 30, and the GCD of 6 and 30 is 6, which satisfies the given information. In this case y = 10
Case b: x = 10 and y = 2. In this case, 2x = 20 and 2y = 4, and the least common multiple of 20 and 4 is 20. Also notice that 3x = 30 and 3y = 6, and the GCD of 30 and 6 is 6, which satisfies the given information. In this case y = 2
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Answer: A
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Re: x and y are positive integers. If the greatest common divisor of 3x and 3y is 6, what is the value of y?

by GMATGuruNY » Sun Aug 16, 2020 11:36 am
BTGModeratorVI wrote: ↑
Fri Aug 14, 2020 1:10 pm
 x and y are positive integers. If the greatest common divisor of 3x and 3y is 6, what is the value of y?

(1) The greatest common divisor of 2x and 2y is 2y
(2) The least common multiple of 2x and 2y is 2
The greatest common divisor of 3x and 3y is 6.
To simplify this statement and isolate x and y, divide every value by 3:
The greatest common divisor of 3x/3 and 3y/3 is 6/3.
In other words:
The greatest common divisor of x and y is 2.

Statement 1:
The greatest common divisor of 2x and 2y is 2y.
To simplify this statement and isolate x and y, divide every value by 2:
The greatest common divisor of 2x/2 and 2y/2 is 2y/2.
In other words:
The greatest common divisor of x and y is y.

Statement 1 indicates that the GCD = y.
The prompt indicates that the GCD = 2.
Thus, y=2.
SUFFICIENT.

Statement 2:
The least common multiple of 2x and 2y is 20
To simplify this statement and isolate x and y, divide every value by 2:
The least common multiple of 2x/2 and 2y/2 is 20/2.
In other words:
The least common multiple of x and y is 10.

Case 1: x=2 and y=10, with the result that the GCD = 2 and the LCM = 10
Case 2: x=10 and y=2, with the result that the GCD = 2 and the LCM = 10
Since y can be different values, INSUFFICIENT.

The correct answer is A.
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