Plug in some actual numbers for z and see if you can answer your own question. If not please return for more help.

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Jamie Nelson
ManhattanGMAT Instructor

Are you sure [1] and [2] are correct?
[1]: 6/2 != EVEN
[2]: 12*3 and 12*6 have GCD 12*3
Am I missing something?

Anyhow, I think, the right thought process for this problem is to substitute the statements' given info into original given info.

Hence, according to Stmt (2),
y = 12z
x = 12 (8z + 1)
Now what? Can we factor anything out of z and 8z+1 to make the GCD 12+ in some way?

Here is the kicker: If A is a number with p1 as "smallest" prime factor, then the "next" number after A (on the increasing number line) that has at-least one factor (other than 1) common to A will be A + p1 away. Eg. 20 has smaller prime factor as 2, so 20+2 is the next number that has a common factor with 20. Note 20+1 will not have any factor common with 21.

Let's get back to the problem. We were at finding common factors b/w z and 8z+1. Smallest prime factor of 8z is 2. Hence, 8z+1 will NOT have any common factor 8z. Since 2 is the smallest PRIME number in-general it can be said that 8z+1 and z will NOT have any common factor. Thus, x and y (above) will ONLY have 12 as GCD. QED!

PS: NOTE:
If we had,
y=12z
x=12 (2z+3)
Then statement (2) would have been insufficient.
Why?
z=1 => GCD=12
z=3 => GCD=36

However, with
y=12z
x=12 (2z+3)
Given z is not a multiple of 3,
It's sufficient.

You can try to imagine more such cases, and doing so should help you exercise your brain along the lines of factors b/w 2 numbers in general.

Thanks for sharing your thoughts. Couldn't tell if there were any non-rhetorical questions in your post, but if there were please ask them again and we'll be glad to help..

_________________
Tim Sanders
Manhattan GMAT Instructor

If x and y are positive integers such that x = 8y + 12, what is the
greatest common divisor of x and y?

sequestered

(1) x = 12u, where u is an integer

y = 3m
u = 2k + 1 = 3, 5, 7, 9, 11

(3, 36) 3
(6, 60) 6
(9, 84) 3
(12, 108) 12
(15, 132) 3
(18, 156) 6
(21, 180) 3
(24, 204) 12

(2) y = 12z, where z is an integer

f(y) = 12z

f(x) = 96z + 12

Values of z greater than 12
are not factors of f(x), since
all factors of f(x) must be
factors of 96z and 12 simultaneously.

Thus, numbers greater than 12 are not common
factors of x and y, and the largest common
factor is known as 12.

Thanks for sharing your thoughts. Couldn't tell if there were any non-rhetorical questions in your post, but if there were please ask them again and we'll be glad to help..

_________________
Tim Sanders
Manhattan GMAT Instructor