To count the factors of a positive integer:

1. Prime-factorize the integer

2. Write the prime-factorization in the form \((a^p)(b^q)(c^r)\)...

3. The number of factors = \((p+1)(q+1)(r+1)\)...

MathRevolution wrote:

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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: \(n=4\)Test one case that also satisfies Statement 2.

Case 1: \(p=5\), with the result that \(3^np^2 = (3^4)(5^2)\)

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 \(3^np^2 = (3^4)(3^2) = 3^6\)

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 \(3^np^2 = (3^2)(5^2)\)

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.

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