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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