x and y are positive integers. If the greatest common divisor of 3x

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:

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Again a very good Q..

x and y are integers and GCD of 3x and 3y is 6..
This means x and y have only one common factor that is 6/3=2..

Let's see the statements

(1) The greatest common divisor of 2x and 2y is 2y
this tells us that x is multiple of y...
But we already know that ONLY common factor between x and y is 2, so y will be 2
Sufficient

(2) The least common multiple of 2x and 2y is 20
LCM of x and y would be 10..
But both x and y are multiples of 2..
So x and y could be 2 and 10, OR 10 and 2
So y can be 2 or 10
Insufficient

A
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Great question as always.


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

Analyzing the prompt:

GCD of 3x and 3y is 6.......it means that both numbers MUST have 2 in its prime factorization, with one of them with power of 1. By definition GCD consists of prime factors with smallest exponent. For example:

3x=12...x=2^2 & 3y=6...y=2...............GCD=6

3x=6.....x=2 & 3y=12..y=2^2..........GCD=6

3x=12...x=2^2 & 3y=18..y=6..............GCD=6

3x=12...x=2^2 & 3y=30...y=10...........GCD=6 (Notice that 30 has 5 so smallest factor is 5^0 in 12)


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

Spotting the statement above means that 2x is dividable by 2y.

2x/2y= 2y.......x=2y^2....... it becomes the following:

GCD of 6y^2 & 3y = 6.......There is one solution which is y=2.

Sufficient

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

20= (2^2) (5)

3x=3*2=6 & 3y=3*2*5=30.........GDC=6.......y=10

3x=3*2*5=30 & 3y=3*2=6...............GCD=6.......y=2

Insufficient

Answer: A

GCF(3x,3y) = 6 => implies that x = 2m, y = 2n, where m and n are co-primes

Statement 1: GCF(2x, 2y) = 2y => which means y is multiple of x, and this can happen only when y = 2 (n = 1). sufficient
Statement 2: LCM(2x,2y) = 20
I can see two cases, x = 2, y = 2 * 5
or x = 2 * 5 , y = 2
not sufficient

(A)

HCF(3x, 3y)=6
therefore HCF(x, y)= 2

Statement 1- HCF of (2x, 2y)= 2y
HCF(x, y)=y
Hence y=2
Sufficient

Statement 2
LCM(2x, 2y)= 20
LCM(x, y)=10
x*y= HCF(x, y)*LCM(x, y)=2*10=20
x and y both are even, hence y can be either 2 or 10
Insufficient

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.


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I was wondering, is this linearity of the GCF/LCM operator valid for every such type of relationship?