Hi, medenica
Are you sure you transcribed the question completely accurately? I think you may have transposed x and y in your original formula. So, you typed "y=8x+12" but this problem is in OG and it is printed as "x = 8y+12" which makes all the difference. As you transcribed it, the statement is insufficient. As the OG prints it, the statement is sufficient.
Also, in future, please do post the whole question, as it is usually better for comprehension to be able to see the entire problem.
So, I'll explain the OG question:
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
A number properties rule we all need to know:
If two integers are added (or subtracted), any factors shared by those two integers will also be factors of the sum (or difference). For example, 2 + 4 = 6. 2 and 4 both have 2 as a factor. 2 is also a factor of 6.
(1) is insufficient. This basically tells us that 12 is a divisor of x. So if we plug into our original equation, we have (multiple of 12) = 8y + (multiple of 12). This tells us that the expression 8y must also be a multiple of 12. The prime factorization of 12 is 2*2*3. The 8 in the expression 8y can cover the two 2's that we need, but not the 3. So the 3 must be a part of y. At the least, then, y is a multiple of 3. x is also a multiple of 3 (since it is a multiple of 12), but this may not be the GREATEST common divisor of the two. Insufficient.
(2) is sufficient. So if we plug into our original equation, we have x = (multiple of 12) + (multiple of 12). Which means that x must also be a multiple of 12. Now, I know that both x and y are multiples of 12, but I still need to address the GREATEST common factor issue.
Number properties rule #2 we need to know:
If one number is
n units away from another number, and
n is a factor of both of those numbers, then the greatest common factor of the two numbers is
n.
This is REALLY complicated. I'll explain this concept below, but you may just want to memorize the above rule. If so, skip to the paragraph that starts "back to our problem." By definition, the greatest common factor divides both numbers evenly without remainder - so the GCF cannot be bigger than the difference between the two numbers.
For example, 24 + 36 = 60. 12 is a factor of all 3 numbers, and the two smaller numbers are 12 units apart. We can re-write this as (12*2) + (12*3) = 60. So 12*3 is 12 units (or 12*1) away from 12*2. The higher number (36, or 12 *
3) is exactly
one multiple higher than the lower number (24, or 12 *
2). We can't get any simpler than one multiple higher, so we can't go any higher than 12 for the GCF.
Back to our problem. So statement 2 tells us that 12 is a factor of both x and 8y, AND it tells us that x is 12 units away from 8y. By definition, then 12 is the GCF of x and 8y. But wait - the problem asks us about x and y, not x and 8y. Well, 8y is eight times bigger than y. So if 12 is the GCF of x and 8y, AND 12 is also a factor of y by itself, then 12 must be the GCF of x and y too. The GCF of x and y can't be bigger than the GCF of x and the
larger multiple 8y.
This is a great one to skip on the test. Just make some kind of educated guess and MOVE ON!!!
