x, y, a, and b are positive integers. When x is divided by y

I would suggest you to analyze each statement as you read it. Often, you will find that you are very close to the answer by the time you read the last sentence of the question.

"When x is divided by y, the remainder is 6."
Here, I say to myself, "Ok, so y must be greater than 6 and x is either 6 or at least 6 greater than y."

If you are not sure how I deduced this, check out these posts:
https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2011/04 ... unraveled/
https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2011/04 ... y-applied/
If y is anything less than 6, you can make one more group!

When a is divided by b, the remainder is 9.
Now I say, "b must be greater than 9 and a is either 9 or at least 9 greater than y."

Which of the following is NOT a possible value for y + b
I already know that y must be greater than 6 and b must be greater than 9. So (y+b) must be greater than 15. Now I will just look for an option <= 15

Question: If 15 is not in the answer choice, is the answer to this question "any number that is not a multiple of 10 + multiple of 7"?

So, 37 and 38 would be okay, but 39 no, correct?

.

What I meant was that the result cannot be something that is a multiple of 10x + 7y. 37 is the same as 30(3) + 7(1). 38 = 7(4) + 10(1). But there is no way to arrive at 39 with a multiple of 10 + multiple of 7. In other words, my question is: Would the answer to this question be "anything that is not "10x + 7y" or is it simply "x + y must be more than 16?"

Doesnt this question have anything to do with multiples of x and y?

Why not? Sum can be 17 or 37 or 38.

It is possible that y = 7 and b = 10.
x = 13, y = 7, remainder of x/y = 6
a = 19, b = 10, remainder of a/b = 9
y + b = 17

Sum 37 is also possible:
It is possible that y = 7 and b = 30.
x = 13, y = 7, remainder of x/y = 6
a = 39, b = 30, remainder of a/b = 9
y + b = 37

etc

All the options except 15 is actually a multiple of 10x + 7y

a)24 - same as 7*2 + 10*1
b)21 - same as 7*3 + 10*0
c)20 - same as 7*0 + 10*2
d)17 - same as 10*1 + 7*1
e)15 - No combination of 10x + 7y can give us 15.


So: Is this just a coincidence here, or is the "pattern" to this question? i.e: Could the question stem just as well say "Which of the following answer choices is not a multiple of 10 + multiple of 7?