If p is a prime number and n is a positive integer, what is the number

To find the number of factors of \(3^n p^2\)

Statement 1

n = 4

If p \(\neq\) 3 then number of factors = (n + 1) * (2 + 1) = 5 * 3 = 15

If p = 3 then number of factors = n + 2 + 1 = 7

Statement 1 is not sufficient

Statement 2

p > 4

number of factors of \(3^n p^2\) = (n + 1) * (2 + 1)

Since we don't know the value of n, statement 2 is insufficient

Combining statements 1 and 2

p > 4 => p \(\neq\) 3 then number of factors = (n + 1) * (2 + 1) = 5 * 3 = 15

Hence option C
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Given: n=4, p=prime number. To know the number of factors, we need to know the powers of primes and whether p=3.

1) If p=3, then the total number of factors= 6+1 =7. But if p is not equal to 3, then the total number of factors will be (4+1)*(2+1) = 5*3 = 15. Not Sufficient.

2) If p>4, then the total number of factors = (2+1)*(n+1). But we do not know the value of n. Hence not sufficient.

Combining 1 & 2, we get n=4 and p is not equal to 3. Therefore the total number of factors = (4+1)*(2+1) = 15. Sufficient. Therefore, C.

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 = \((p+1)(q+1)(r+1)\)...



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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Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.

If a question asks for a number of factors, it is very important to check that all of the given prime factors are “different”. By condition 2), \(p\) is a prime number different from \(3\). To determine the number of factors, we need to know the exponents in the prime number factorization. Therefore, we also need condition 1).

Since \(p\) is a different prime integer from \(3\), and \(n = 4\), the number of factors of \(3^np^2\) is \((4+1)(2+1) = 15.\)
Since we have a unique solution, both conditions together are sufficient.

Therefore, C is the answer.
Answer: C
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Solution:

Given: ‘p’ is a prime number and ‘n’ is a positive integer.
To find: The number of factors of \(3^n p^2\)?
Inferences: To find the number of factors; the best technique is “prime factorization”. Any composite number can be represented in the form of its prime factors. We can represent as \((a^p)(b^q )(c^r )\).
Then the number of factors we can find \((p+1)(q+1)(r+1)\).

Analysis of statement 1: \(n=4\)
We can get two cases here,
Case A: Let the value of \(p = 7\),
Then total number of factors = \(3^4 7^2\)= (4 + 1) (2 + 1) = (5)(3) = 15 factors.
Case B: Let the value of \(p = 3\)
Then total number of factors = \(3^4 3^2\)= \(3^6\) = (6 + 1) = 7 factors.
Here we are getting two different answers; hence the statement 1 is not sufficient. We can eliminate the options A and D.

Analysis of statement 2: \(p > 4\)
Here too we can have two cases,
Case A: Let the value of \(p = 7\),
Then total number of factors = \(3^4 7^2\)= (4 + 1) (2 + 1) = (5)(3) = 15 factors.
Case A also satisfies statement 2.
Case C: \(p = 7\) and \(n = 2\)
The total number of factors = \(3^2 7^2\)=(2+1)(2+1)= (3)(3) = 9 factors.
Here we are getting two different answers; hence the statement 2 is not sufficient. We can eliminate the option B.

Combining the statements 1 and 2; we get:
When \(n = 4\) and \(p > 4\), the number of factors = 15, as encapsulated in case A.
Hence sufficient.

The correct answer option is “C”.

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