To count the factors of a positive integer:
1. Prime-factorize the integer
2. Write the prime-factorization in the form [m](a^p)(b^q)(c^r)[/m]...
3. The number of factors = [m](p+1)(q+1)(r+1)[/m]...
MathRevolution wrote:[Math Revolution GMAT math practice question]
If p is a prime number and n is a positive integer, what is the number of factors of 3^np^2?
1) n = 4
2) p > 4
Statement 1: [m]n=4[/m]
Test one case that also satisfies Statement 2.
Case 1: p=5, with the result that (3^n)(p^2) = 3�5²
Adding 1 to each exponent and multiplying, we get:
Number of factors = (4+1)(2+1) = 15
Test a case that does NOT also satisfy Statement 2.
Case 1: p=3, with the result that (3^n)(p^2) = 3�3² = 3�
Adding 1 to the only exponent, we get:
Number of factors = 6+1 = 7
Since the number of factors can be different values, INSUFFICIENT.
Statement 2: p>4
Case 1 also satisfies Statement 2.
In Case 1, the number of factors = 15.
Case 3: p=5 and n=2, with the result that (3^n)(p^2) = 3²5²
Adding 1 to each exponent and multiplying, we get:
Number of factors = (2+1)(2+1) = 9
Since the number of factors can be different values, INSUFFICIENT.
Statements combined:
As illustrated by Case 1, if n=4 and p>4, the number of factors = 15.
SUFFICIENT.
The correct answer is
C.
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