j_shreyans 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 = a²b³, where a and b are different prime numbers
(2) b=a+1

Target question: 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?

This target question is really just asking, "How many positive divisors does x have?"
For example, if x = 12 (i.e., there are 12 biscuits), how many different values of y (i.e., the # of patrons) will be such that y divides into x?
Well, if y = 1, 2, 3, 4, 6 or 12 (all divisors of 12), we can meet the requirement that each patron receives an equal number of whole biscuits and with no biscuits left over.

So, let's REPHRASE the target question:
REPHRASED target question: How many positive divisors does x have?

Aside: We have a free video with tips on rephrasing the target question: https://www.gmatprepnow.com/module/gmat- ... cy?id=1100

Statement 1: x = a²b³, where a and b are different prime numbers
There's a nice rule that says: If the prime factorization of N = (p^a)(q^b)(r^c) . . . (where p, q, r, etc are different prime numbers), then N has a total of (a+1)(b+1)(c+1)(etc) positive divisors.

Example: 14000 = (2^4)(5^3)(7^1)
So, the number of positive divisors of 14000 = (4+1)(3+1)(1+1) =(5)(4)(2) = 40

So, if a²b³ (where a and b are different prime numbers), then the number of positive divisors of x = (2+1)(3+1) = (3)(4) = 12
So, x has 12 positive divisors
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: b = a+1
At this point (if we ignore statement 1), we don't even know what a and b represent.
So, statement 2 is NOT SUFFICIENT

Answer = A

Cheers,
Brent

Hi j_shreyans,

Many DS questions are built on "math patterns" that you might not immediately recognize. If you're not sure about whether a given Fact is Sufficient or Insufficient, then you have to TEST Values a few times to see if a pattern emerges.

Here, we're told that we're going to evenly distribute X biscuits to Y patrols, with no biscuits left over. Since the number of patrons can vary, we're asked to determine how many variations are possible (based on the number of biscuits). This is essentially about multiples and factors, but you don't need to know that to answer the question.

Fact 1: X = A²B³, A and B are different primes

Many people would look at this and think it was insufficient, but let's PROVE what's really going on here by TESTing Values.

If...
A = 3
B = 2
X = 72
72 biscuits could be evenly distributed to what number(s) of people?
1 & 72, 2 & 36, 3 & 24, 4 & 18, 6 & 12, 8 & 9 --> 12 possible values for Y

A = 2
B = 3
X = 108
108 biscuits could be evenly distributed to what number(s) of people?
1 & 108, 2 & 54, 3 & 36, 4 & 27, 6 & 18, 9 & 12 --> 12 possible values of Y

That's pretty interesting that the answer is EXACTLY THE SAME. I'm going to run one more TEST just to be sure...

A = 5
B = 2
X = 200
200 biscuits could be evenly distributed to what number(s) of people?
1 & 200, 2 & 100, 4 & 50, 5 & 40, 8 & 25, 10 & 20 --> 12 possible values of Y

At this point, I'm convinced that the answer is ALWAYS 12.
Fact 1 is SUFFICIENT.

Fact 2: B = A + 1

On it's own, this Fact has no reference/relevance to the prompt.
Fact 2 is INSUFFICIENT

Final Answer: A

GMAT assassins aren't born, they're made,
Rich