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Number properties, divisibility, GCF

Problem Solving — algebra and arithmetic (GMAT Focus Edition)
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Number properties, divisibility, GCF

by RSK » Mon Feb 20, 2012 8:31 am
Hi guys,

Is there a quick way to solve this question? The method I used involved a lot of work and time.

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.
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by kul512 » Thu Feb 23, 2012 7:40 am
Given Information-
X=8Y+12
X=4(2Y+3)
From this information we know that X is at least multiple of 4, now we need some information about Y to get the Greatest common multiple of X and Y.

Statement 1:
X=12U
This statement tells that X is multiple of 12. So 2Y+3 must have a factor of 3, or Y must be multiple of 3. Many values of Y (3,6,9 ........Multiple of three) can be derived from this information. And in each case GCD is a different value. So this information is insufficient.

Statement 2:
Y=12Z If we put this value of Y in the main information i.e.X=8Y+12, then
X=12(8Z+1)
so both X and Y have a common divisor of 12. Now is it the greatest divisor for X and Y, yes. Why?
Because 8Z+1 and Z cannot have a common factor. So from here we get our answer that Y and X has a greatest common divisor of 12. So sufficient

Answer (B)
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by krusta80 » Tue Feb 28, 2012 9:21 pm
RSK wrote:Hi guys,

Is there a quick way to solve this question? The method I used involved a lot of work and time.

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.
If y = 1, then 1 is the gcd. INSUFF
If y = 2, then 2 is the gcd. INSUFF


(1) x mod 12 = 0

Since x = 8y + 12, we have:

(8y + 12) mod 12 = 0
8y mod 12 = 0

y mod 3 = 0

If y = 3, then GCD is 3
If y = 6, then GCD is 6

INSUFF


(2) y mod 12 = 0

(8y+12) mod 12 = 0

Since y is multiplied by 8, the result will always be an even coefficient times 12. To get x, 12 is added to 8y, which results in an odd coefficient of 12.

Therefore, 12 must by the GCD.

SUFFICIENT => B
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by RSK » Tue Feb 28, 2012 10:38 pm
Sorry I don't understand the solution. What do you mean by 'mod'?
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by krusta80 » Wed Feb 29, 2012 6:18 am
RSK wrote:Sorry I don't understand the solution. What do you mean by 'mod'?
Mod means remainder when divided by.

So, 6 mod 5 means the remainder of 6 divided by 5, which of course is 1.
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