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x and y are positive integers. If the greatest common divisor of 2x

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BrentGMATPrepNow
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x and y are positive integers. If the greatest common divisor of 2x [#permalink] New post   03 Apr 2017, 07:43
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x and y are positive integers. If the greatest common divisor of 2x and 2y is 30, what is the greatest common divisor of x and 2y?

1) y is odd
2) x is odd
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BrentGMATPrepNow
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x and y are positive integers. If the greatest common divisor of 2x [#permalink] New post   Updated on: 02 Mar 2020, 07:14
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GMATPrepNow wrote:
x and y are positive integers. If the greatest common divisor of 2x and 2y is 30, what is the greatest common divisor of x and 2y?

1) y is odd
2) x is odd


Target question: What is the greatest common divisor of x and 2y?

Given: the greatest common divisor of 2x and 2y is 30
30 = (2)(3)(5)
This means that, if we examine the prime factorization of 2x and prime factorization of 2y, they will share exactly ONE 2, ONE 3, and ONE 5.

That is:
2x = (2)(3)(5)(?)(?)(?)
2y = (2)(3)(5)(?)(?)(?)

NOTE: Both prime factorizations might include other primes, BUT there is no additional overlap beyond the ONE 2, ONE 3, and ONE 5.

Notice that if we divide both sides of both prime factorizations by 2, we get:
x = (3)(5)(?)(?)(?)
y = (3)(5)(?)(?)(?)
Since we already know that there is no additional overlap beyond the ONE 3, and ONE 5, we can conclude that the greatest common divisor (GCD) of x and y is 15.

Since we're trying to find the greatest common divisor of x and 2y, we should take a closer look at the prime factorizations of x and 2y:
x = (3)(5)(?)(?)(?)
2y = (2)(3)(5)(?)(?)(?)

We already know that x and y have no additional overlap beyond the ONE 3, and ONE 5, the GCD of x and 2y will be EITHER 15 OR 30

If the prime factorization of x contains a 2, then x and 2y will share ONE 2, ONE 3, and ONE 5, which means the GCD of x and 2y will be 30

If the prime factorization of x does not contain a 2, then x and 2y will share ONE 3, and ONE 5, which means the GCD of x and 2y will be 15

So, it all comes down to whether or not the prime factorization of x contains a 2.


Statement 1: y is odd
This information does not tell us whether or not the prime factorization of x contains a 2
There are several values of x and y that satisfy statement 1. Here are two:
Case a: x = 15 and y = 15. This satisfies the given condition that the GCD of 2x and 2y is 30. In this case the GCD of x and 2y is 15
Case b: x = 30 and y = 15. This satisfies the given condition that the GCD of 2x and 2y is 30. In this case the GCD of x and 2y is 30
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: x is odd
If x is ODD, then we know that the prime factorization of x does not contain a 2, which means the GCD of x and 2y is 15
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer: B

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Originally posted by BrentGMATPrepNow on 04 Apr 2017, 08:31.
Last edited by BrentGMATPrepNow on 02 Mar 2020, 07:14, edited 1 time in total.
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Re: x and y are positive integers. If the greatest common divisor of 2x [#permalink] New post   03 Apr 2017, 11:06
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As per the information given in the question we can write 2x and 2y as below:

2 * x = 2 * 3 * 5 * a
2 * y = 2 * 3 * 5 * b

a and b above represent any possible factors of 2x and 2y, where a and b do not have any common prime factor

Statement 1) y is odd


If y is odd, we know that b is also odd. Now If a in 2*x is also odd, then the greatest common divisor of x and 2y is 3*5, but if a in 2*x is even, then the greatest common divisor of x and 2y is 2*3*5. As we have two possibilities, hence this statement is not sufficient.


Statement 2) x is odd

If x is odd, we know that a is also odd.

This means that x can be written as x = 3 * 5 * a (Where a is odd)

and 2 * y = 2 * 3 * 5 * b

As from the information given we already know a and b do not have any common prime factors, the greatest common divisor of x and 2y is 3*5 = 15

This statement is sufficient.

Answer is B
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Re: x and y are positive integers. If the greatest common divisor of 2x [#permalink] New post   03 Apr 2017, 11:23
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GMATPrepNow wrote:
x and y are positive integers. If the greatest common divisor of 2x and 2y is 30, what is the greatest common divisor of x and 2y?

1) y is odd
2) x is odd



(10 y=30, x= 15 gcd=15
but
y=15 , x=60 gcd =30
insuff

(2)x=15* m(odd) then 2Y=even* odd(since 2x & 2y have gcd =15)
thus x *2y have gcd =15
suff

Ans B
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x and y are positive integers. If the greatest common divisor of 2x [#permalink] New post   Updated on: 11 Apr 2024, 22:39
we are given that the GCF of (2x and 2y) = 30 = (2) (3) (5)

The implication of this fact is that the ONLY Prime Factors that 2X and 2Y can share are: 2, 3 , 5 ---- all to the 1st Power

2X = (2) (3) (5) * (A)

2Y = (2) (3) (5) * (B)

where A and B are the remaining factors of 2X and 2Y, respectively, and where A and B are COPRIME.

This means that A and B can NOT SHARE ANY FACTORS Except 1.

GCF (A and B) = 1

Question asks: What is the GCF of (X and 2Y) = ?

X = (3) (5) * (A)

2Y = (2) (3) (5) * (B)

we already know that A and B can share NO PRIME FACTORS other than 1

so there are ONLY TWO Possible Answers:

CASE 1:
X contains at least ONE 2 Prime Factor
In other words: A = (2) (……)

This means:

X = (3) (5) * (A) = (3) (5) * (2 * ..........)

2Y = (2) (3) (5) * (B) = (2) (3) (5) * (.......any prime factors OTHER THAN what is in A…..)

hence, GCF must be the only shared primes of: (2) (3) (5) = 30

CASE 2:
the remaining factors of X are ODD, making X an ODD Number
In other words, Adoes NOT contain any 2 prime factors

X = (3)(5) * (A) = (3) (5) * (Odd primes)

2Y = (3)(5) * (B) = (2) (3) (5) * (...prime factors that are DIFFERENT FROM what is in A.....)

the GCF in this case would be the only shared Primes: (3) (5) = 15

QUESTION THEN BECOMES:
is the GCF [X and 2Y] = 15 or 30?
OR
Does X contain any 2 prime factors?

s1: Y = ODD

case 1 or case 2 is possible. the GCF could be either 15 or 30.
s1 not sufficient.

s2: X = ODD

statement 2 tells us that:

X = (3)(5) * (A)

the remaining factors encompassed in (A) can NOT contain any prime factors of 2. Otherwise, X would be Even and NOT Odd.

we know that 2Y is:
2Y = (2) (3) (5) * (B)

where A and B do NOT share any Prime Factors

therefore the only Prime Factors that X and 2Y do not include any numbers from A or B and the GCF must be: (3) (5) = 15

s2 sufficient alone --- GCF = 15

*B*

Originally posted by Fdambro294 on 17 Nov 2022, 23:20.
Last edited by Fdambro294 on 11 Apr 2024, 22:39, edited 1 time in total.
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Re: x and y are positive integers. If the greatest common divisor of 2x [#permalink] New post   28 Mar 2024, 10:01
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Re: x and y are positive integers. If the greatest common divisor of 2x [#permalink]
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