MathRevolution wrote:
[GMAT math practice question]
If \(x\) and \(y\) are prime numbers, and \(n\) is a positive integer, what is the number of factors of \(x^ny^n\)?
\(1) xy=6\)
\(2) n=2\)
-------ASIDE----------------
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
-----NOW ONTO THE QUESTION----------------------
Target question: What is the number of factors of (x^n)(y^n)? Statement 1: xy =6 There are several values of x, y and n that satisfy statement 1. Here are two:
Case a: x = 2, y = 3 and n = 1. We get the expression: (2^
1)(3^
1). So, the number of positive divisors = (
1 + 1)(
1 + 1) = 4. So, the answer to the target question is
there are 4 divisorsCase b: x = 2, y = 3 and n = 7. We get the expression: (2^
7)(3^
7). So, the number of positive divisors = (
7 + 1)(
7 + 1) = 64. So, the answer to the target question is
there are 64 divisorsSince we cannot answer the
target question with certainty, statement 1 is NOT SUFFICIENT
Statement 2: n=2 There are several values of x, y and n that satisfy statement 2. Here are two:
Case a: x = 2, y = 3 and n = 2. We get the expression: (2^
2)(3^
2). So, the number of positive divisors = (
2 + 1)(
2 + 1) = 9. So, the answer to the target question is
there are 9 divisorsCase b: x = 3, y = 3 and n = 2. We get the expression (3^2)(3^2), which simplifies to be (3^
4). So, the number of positive divisors = (
4 + 1) = 5. So, the answer to the target question is
there are 5 divisorsSince we cannot answer the
target question with certainty, statement 2 is NOT SUFFICIENT
Statements 1 and 2 combined Statement 1 tells us that one number is 2 and the other number is 3
Statement 2 tells us that n = 2
We get the expression: (2^
2)(3^
2). So, the number of positive divisors = (
2 + 1)(
2 + 1) = 9. So, the answer to the target question is
there are 9 divisorsSince we can answer the
target question with certainty, the combined statements are SUFFICIENT
Answer: C
Cheers,
Brent
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