求教PREP数学题

KaedeChou

求解题思路 非常感谢
DS:
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.


Prep Answer B

素心如兰

This is a VERY difficult problem. The vast, vast majority of people will not be able to tackle this problem in 2 minutes and will need to make an educated guess and move on.

Statement (1) tells us that 12 is a divisor of x. What does it tell us about y?
12u = 8y + 12
(multiple of 12) = 8y + (multiple of 12)
8y must be a multiple of 12.

Therefore, y must be a multiple of 3 and 3 is a divisor of y. 3 might be the greatest common divisor of x and y. But y might have other divisors too (e.g., 6 or 12). Insufficient.

Statement (2) tells us that 12 is a divisor of y.
What does it tell us about x?
x = 8(12z) + 12
x = (multiple of 12) + (multiple of 12)
x must be a multiple of 12.
12 is a divisor of x.

So 12 is a common divisor of x and y. But is it the greatest common divisor?

RULE: If one number is b units away from another number, and b is a factor of both numbers, the greatest common factor of the two numbers is b. (If you want to really understand this, then think about why. Otherwise, just remember the rule.)

x (one number) is 12 units away from 8y (another number). 12 is a factor of x and 8y. Therefore, 12 is the GCF of x and 8y. The GCF of x and y can’t be bigger than the GCF of x and 8y. Thus, we can be assured that 12 is the GCF of x and y. Statement (2) alone is SUFFICIENT.
Stacey Koprince
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ManhattanGMAT

KaedeChou

素心如兰 发表于 2015-9-8 22:15
This is a VERY difficult problem. The vast, vast majority of people will not be able to tackle this  ...

Thank you so much!

包子噢

素心如兰 发表于 2015-9-8 22:15
This is a VERY difficult problem. The vast, vast majority of people will not be able to tackle this  ...

同意！