[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
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If p is a prime number and n is a positive integer, what is
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Max@Math Revolution
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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]...
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.
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]...
Statement 1: [m]n=4[/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
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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Max@Math Revolution
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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
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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