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akash singhal: Master | Next Rank: 500 Posts; Posts: 152; Joined: Fri Apr 24, 2015 1:39 amFollowed by: 3 members; Location: Rourkela Odisha India; Thanked: 2 times; GMAT Score:650

by akash singhal » Sat Nov 07, 2015 4:01 am

x and y are positive integers such that x=8y+12, what is the greatest common divisor of x and y?

1). X=12u, where u is an integer. 2). Y=12z, where z is an integer.

OE B

Explain

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Source: Beat The GMAT — Data Sufficiency | Sat Nov 07, 2015 4:01 am

GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Sat Nov 07, 2015 4:12 am

If x and y are positive integers such that x = 8y + 12, what is the greatest common divisor of x and y?

(1) x = 12u, where u is an integer
(2) y = 12z, where z is an integer

Statement 1: x=12u, where u is an integer and x=8y+12.
In other words, x is a multiple of 12.
For x to be a multiple of 12, 8y must be a multiple of 12.

If y=3, then x = 8*3 + 12 = 36.
The GCD of 3 and 36 is 3.

If y=6, then x = 8*6 + 12 = 60.
The GCD of 6 and 60 is 6.

Since the GCD can be different values, INSUFFICIENT.

Statement 2: y=12z, where z is an integer and x=8y+12.
In other words, y is a multiple of 12.
Since we're looking for the GCD, view x in terms of its FACTORS.

If y=12, then x = 8(12) + 12 = 12(8+1) = 12*9.
The GCD of 12 and 12*9 is 12.

If y=24, then x = 8(24) + 12 = 12(8*2 + 1) = 12*17.
The GCD of 24 and 12*17 is 12.

I'm almost convinced: the GCD is 12.
Maybe one more just to be sure:

If y=36, then x = 8(36) + 12 = 12(8*3 + 1) = 12*25.
The GCD of 36 and 12*25 is 12.
SUFFICIENT.

The correct answer is B.

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GMATinsight: Legendary Member; Posts: 1100; Joined: Sat May 10, 2014 11:34 pmFollowed by: 24 members; Location: New Delhi, India; Thanked: 205 times

by GMATinsight » Sat Nov 07, 2015 5:59 am

akash singhal wrote:x and y are positive integers such that x=8y+12, what is the greatest common divisor of x and y?

1). X=12u, where u is an integer.
2). Y=12z, where z is an integer.

OE B

Explain

x=8y+12 = 4(2y+3)
i.e. x is a Multiple of 4

Statement 1: X=12u

i.e. x is a multiple of 12
i.e. y must be a multiple of 3
but since y may be an even multiple of 3 or an odd multiple of 3 so GCD will have different values. Hence,
NOT SUFFICIENT

Statement 2: Y=12z

i.e. y must be a multiple of 3 as well 4
for such value of y, x must be a multiple of 12
e.g. @y=12, x = 4*27, GCD = 12
@y=24, x = 4*51, GCD = 12
but since y is an even multiple of 3 so GCD will have constant value. Hence,
SUFFICIENT

Answer: Option B

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Sat Nov 07, 2015 7:36 am

If x and y are positive integers such that x = 8y + 12, what is the greatest common divisor of x and y?

(1) x = 12u, where u is an integer
(2) y = 12z, where z is an integer

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

Given: x = 8y + 12

Statement 1: x = 12u, where u is an integer.
There are several pairs of values that satisfy the given conditions. Here are two:
Case a: x=36 and y=3, in which case the GCD of x and y is 3
Case b: x=60 and y=6, in which case the GCD of x and y is 6
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: y = 12z, where z is an integer.
If y = 12z and x = 8y + 12, then we can replace y with 12z to get:
x = 8(12z) + 12, which means x = 96z + 12, which means x = 12(8z + 1) [if we factor]

So, what is the GCD of 12z and 12(8z + 1)?
Well, we can see that they both share 12 as a common divisor, but what about z and 8z+1?
Well, there's a nice rule that says: The GCD of n and kn+1 is always 1 (if n and k are positive integers)
So, the GCD of z and 8z+1 is 1, which means the GCD of 12z and 12(8z + 1) is 12.
This means that the GCD of x and y is 12
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer = B

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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Amrabdelnaby: Master | Next Rank: 500 Posts; Posts: 137; Joined: Fri Nov 13, 2015 11:01 amFollowed by: 2 members; Thanked: 1 times

by Amrabdelnaby » Sun Nov 22, 2015 6:16 pm

Hi Brent,

I actually solved it using algebra. Pls tell me if my thread of thoughts is right.

Statement 1: x = 12U
12U = 8y + 12 ---> divide by 4
3U = 2y + 3

Insufficient because we have 2 unknown variables and we can't get a common factor for them

Statement 2: y = 12z

x = 8(12z) + 12
x = 12(8z + 1)

here the highest common factor is 12 so this statement is sufficient.

Is this correct thinking?

GMATGuruNY wrote:
If x and y are positive integers such that x = 8y + 12, what is the greatest common divisor of x and y?

(1) x = 12u, where u is an integer
(2) y = 12z, where z is an integer
Statement 1: x=12u, where u is an integer and x=8y+12.
In other words, x is a multiple of 12.
For x to be a multiple of 12, 8y must be a multiple of 12.

If y=3, then x = 8*3 + 12 = 36.
The GCD of 3 and 36 is 3.

If y=6, then x = 8*6 + 12 = 60.
The GCD of 6 and 60 is 6.

Since the GCD can be different values, INSUFFICIENT.

Statement 2: y=12z, where z is an integer and x=8y+12.
In other words, y is a multiple of 12.
Since we're looking for the GCD, view x in terms of its FACTORS.

If y=12, then x = 8(12) + 12 = 12(8+1) = 12*9.
The GCD of 12 and 12*9 is 12.

If y=24, then x = 8(24) + 12 = 12(8*2 + 1) = 12*17.
The GCD of 24 and 12*17 is 12.

I'm almost convinced: the GCD is 12.
Maybe one more just to be sure:

If y=36, then x = 8(36) + 12 = 12(8*3 + 1) = 12*25.
The GCD of 36 and 12*25 is 12.
SUFFICIENT.

The correct answer is B.

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Matt@VeritasPrep: GMAT Instructor; Posts: 2630; Joined: Wed Sep 12, 2012 3:32 pmFollowed by: 119 members; Location: East Bay all the way; Thanked: 625 times; GMAT Score:780

by Matt@VeritasPrep » Fri Nov 27, 2015 1:27 am

Amrabdelnaby wrote:Hi Brent,

I actually solved it using algebra. Pls tell me if my thread of thoughts is right.

Statement 1: x = 12U
12U = 8y + 12 ---> divide by 4
3U = 2y + 3

Insufficient because we have 2 unknown variables and we can't get a common factor for them

Statement 2: y = 12z

x = 8(12z) + 12
x = 12(8z + 1)

here the highest common factor is 12 so this statement is sufficient.

Is this correct thinking?

The logic applied to the first statement is dangerous. For instance, suppose that I tell you that p and q are distinct primes, and that m = 2p and n = 4q. From this, we know that the GCF of m and n = 2, so we could solve a similar problem.

"Two variables, two equations" is one of the more dangerous justifications to use on any GMAT problem: the test writers know that students overapply and/or misunderstand this and design many problems to trap you. Know what it does (= it tells you that, given two independent linear equations in x and y, you can uniquely determine x and y) and what it doesn't (= a zillion other things).

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Matt@VeritasPrep: GMAT Instructor; Posts: 2630; Joined: Wed Sep 12, 2012 3:32 pmFollowed by: 119 members; Location: East Bay all the way; Thanked: 625 times; GMAT Score:780

by Matt@VeritasPrep » Fri Nov 27, 2015 1:31 am

That said, we could use algebra to approach the first statement.

Since x = 8y + 12, we know that GCF(x,y) is the same as GCF(8y+12, y). Since the GCF is a factor of y, it must be a factor of 8y. Since it's a factor of 8y and of 8y + 12, it must also be a factor of their difference, or 12. So whatever the GCF is, it's a factor of 12. But the value of x itself doesn't tell us which factor of 12 this must be, so we can't solve.

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