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 a
²b
³ (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
