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Greatest Common Divisor problem

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Greatest Common Divisor problem

by Sent » Sun May 08, 2016 1:04 pm
Ran into a problem in a practice test that I didn't know how to do, so ended up guessing. I feel like I'm not as up-to-scratch when it comes to greatest common divisors.

Question:

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

Answer: B

Any help on how to conceptualize this would be appreciated!
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by Brent@GMATPrepNow » Sun May 08, 2016 1:15 pm
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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