BTGmoderatorDC wrote:At a particular moment, a restaurant has x biscuits and y patron(s), with x>2 and y>1. How many values of y are there, such that all the biscuits can be distributed among the patrons, with each patron receiving an equal number of whole biscuits and with no biscuits left over?

(1) x=a^2*b^3, where a and b are different prime numbers
(2) b=a+1

OA A

Source: Manhattan Prep

So, in a nutshell, we have to find out the possible values of x/y such that x/y is an integer, x > 2 and y > 1.

Let's take each statement one by one.

(1) x = a^2*b^3, where a and b are different prime numbers

Given that x = a^2*b^3, the number of factors of x are (2 + 1)*(3 + 1) = 12. Since 12 factors of x also contains 1, the qualified values of y would be 12 - 1 = 11 as y > 1.

Let's take an example:

x = 2^2*3^3 = 108

Thus, 2^2*3^3 = 108 has (2 + 1)*(3 + 1) = 12. The factors are 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, and 108. for x/y to be an integer, y can take and of the 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, and 108 values, i.e. 11 values. Sufficient.

Don't get confused with 2 as a factor of x, given that x > 2. Given x = a^2*b^3, where a and b are different prime numbers, the minimum value of x = 108 > 2.

Even if you take any other example, you will get the same answer. You may try with x = 5^2*7^3. The number of factors of x would be 12.

(2) b = a + 1

Certainly insufficient.

The correct answer: A

Hope this helps!

-Jay
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It makes no sense to ask a DS question in this way. If y is a "number of patrons" in a real GMAT DS question, then y stands for a single unknown value; it doesn't stand for a variable. You can't ask "how many values of y are possible?" in DS, because it's not clear what information would be sufficient to answer that kind of question. For example, if you saw this question:

If y is a positive integer, how many values of y are possible?
1. y < 5
2. 3 < y < 7

what is the correct answer? Using Statement 1, four values are possible, using Statement 2, three values are possible, and using both statements together, one value is possible. So what information is sufficient? There's no way to tell, because asking a question in this way makes no logical sense. You can never see a question like this, or like the one in the OP above, on the real GMAT.