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Greatest Common Divisor

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by dlencz » Mon Apr 23, 2012 10:58 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

OA is B
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by Shalabh's Quants » Mon Apr 23, 2012 11:21 am
dlencz 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

OA is B
Pl. see here.

https://www.beatthegmat.com/mba/2010/10/ ... ty-problem
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by GMATGuruNY » Mon Apr 23, 2012 7:27 pm
dlencz 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

OA is B
For many test-takers, the most efficient approach on test day would be to plug in.

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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by amit28it » Tue Apr 24, 2012 4:52 am
The solution provided by GMATGuruNY is perfect there is no better solution for this problem,I have just completed the problem and follows the same procedure and the answer is Correct that is B.
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by amit28it » Tue Apr 24, 2012 5:24 am
The solution provided by GMATGuruNY is the simplest solution of this question,I have just completed the question and found that the procedure I am following is same and so is the answer.
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by shantanu86 » Tue Apr 24, 2012 6:21 am
dlencz 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

OA is B
In general if a equation has the form -

x = n*y +c

It automatically implies that the HCF of x & y has to be a factor of 'c' resulting in
x' * HCF = y'*HCF +c'*HCF.. here x', y' and c' are quotient when HCF divides x, y and c respectively.

Thus, 12 will become HCF iff y has the form y= 12z.
Hence B alone is sufficient.

A implies that y =12(u-1)/8 = 4*(u-1)/3.. tells nothing.

Hope it helps!!
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by RBBmba@2014 » Tue Mar 24, 2015 10:32 pm
GMATGuruNY wrote:
dlencz 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

OA is B
For many test-takers, the most efficient approach on test day would be to plug in.

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.
Hi Mitch - your above explanation is based on the following logic, I think.

1. multiple of X +/- multiple of X = multiple of X ALWAYS

2. multiple of X +/- multiple of Y = multiple of Unknown ALWAYS

and these two will always hold good!

Correct me please if wrong.
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by Brent@GMATPrepNow » Wed Mar 25, 2015 7:33 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


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by RBBmba@2014 » Fri Mar 27, 2015 3:05 am
RBBmba@2014 wrote:
GMATGuruNY wrote:
dlencz 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

OA is B
For many test-takers, the most efficient approach on test day would be to plug in.

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.
Hi Mitch - your above explanation is based on the following logic, I think.

1. multiple of X +/- multiple of X = multiple of X ALWAYS

2. multiple of X +/- multiple of Y = multiple of Unknown ALWAYS

and these two will always hold good!

Correct me please if wrong.
Hi Experts - can you please share your thoughts on my above post ?

@Mitch - could you please clarify this Sir!
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by GMATGuruNY » Fri Mar 27, 2015 4:43 am
RBBmba@2014 wrote:
1. multiple of X +/- multiple of X = multiple of X ALWAYS

2. multiple of X +/- multiple of Y = multiple of Unknown ALWAYS
Rule 1 is correct.
I would state Rule 2 as follows:
(multiple of X) ± (non-multiple of X) = (non-multiple of X).
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As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.

For more information, please email me (Mitch Hunt) at [email protected].
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