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Greatest Common Factor Difficult Problem

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by beweezy » Wed Feb 22, 2012 11:09 am
Could anyone help me with this one?

X & Y are positive integers. X=8y+12. What is the GCF of X & Y?

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

I believe the answer is (B) but I'm not positive. Thanks!
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Source: — Data Sufficiency |
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by GMATGuruNY » Wed Feb 22, 2012 6:51 pm
beweezy wrote:Could anyone help me with this one?

X & Y are positive integers. X=8y+12. What is the GCF of X & Y?

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

I believe the answer is (B) but I'm not positive. Thanks!
Statement 1: x = 12u, where u is an integer.
Substituting x=12u into x = 8y + 12, we get:
12u = 8y + 12.
u = (2/3)y + 1.

Since u is an integer, y must be a multiple of 3.
Since x=12u, x is a multiple of 12.
If y=3, the GCF of x and y is 3.
If y=6, the GCF of x and y is 6.
Since the GCF can be different values, INSUFFICIENT.

Statement 2: y = 12z, where z is an integer.
Substituting y=12z into x = 8y + 12, we get:
x = 8(12z) + 12.
x = 12(8z + 1).

Factors of x are 12 and 8z+1.
Factors of y are 12 and z.
12 is a factor of x and y.
The only other possible GCF is z.
In order for z to be the GCF, it would have to be greater than 12 and a factor of 8z+1.
But since z is a factor of 8z, it is NOT a factor of 8z+1.
The reason is that 8z and 8z+1 are consecutive integers, and consecutive integers are COPRIMES; they share no factors other than 1.
Thus, the GCF of x and y is 12.
SUFFICIENT.

The correct answer is B.
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by beweezy » Thu Feb 23, 2012 3:53 am
Wow - that's rough. Thanks!
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by GMATGuruNY » Thu Feb 23, 2012 4:12 am
beweezy wrote:Wow - that's rough. Thanks!
An less abstract approach would be to plug in values.

Statement 1: x=12u, where u is an integer.
In other words, x is a multiple of 12.
If y=3, then x=8(3)+12 = 36.
The GCF of 3 and 36 is 3.
If y=6, then x=8(6)+12 = 60.
The GCF of 6 and 60 is 6.
Since the GCF can be different values, INSUFFICIENT.

Statement 2: y=12z, where z is an integer.
In other words, y is a multiple of 12.
If y=12, then x=8(12)+12 = 12(8+1) = 12*9.
The GCF of 12 and 12*9 is 12.
If y=24, then x=8(24)+12 = 12(16+1) = 12*17.
The GCF of 24 and 12*17 is 12.
At this point, we can be pretty assured that the GCF is 12.
One more just to check:
If y=36, then x=8(36)+12 = 12(24+1) = 12*25.
The GCF of 36 and 12*25 is 12.
In every case, the GCF is 12.
SUFFICIENT.

The correct answer is B.
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