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How many even divisors?

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How many even divisors?

by LulaBrazilia » Fri Mar 07, 2014 10:53 am
If n=4p, where p is a prime number greater than 2, how many different positive even divisors does n have, including n?

A) 2
B) 3
C) 4
D) 6
E) 8

Is it safe to plug in? Is there a quick algebraic approach?
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by Patrick_GMATFix » Fri Mar 07, 2014 11:02 am
Lula,

Solution A

In general, the number of positive factors can be calculated from the prime factorization. If x = m^a * n^b * p^c... where m,n and p are the distinct prime factors of x, then the number of factors of x is (a+1)*(b+1)*(c+1)... (the values of m,n,p... don't matter).

For instance, 45 = 3^2 * 5^1, so it must have (2+1)*(1+1) = 6 positive factors. They are {1, 3, 5, 9, 15, and 45}

In this case, n=2^2 * k^1.

Number of factors: (2+1)*(1+1) = 6 factors.
Odd factors: {1, k} = 2 odd factors.
Even factors = 6-2 = 4 even factors.

The answer is C.


Solution B

Since all values of n must result in the same answer, plugging in an easy value for n is definitely what I would opt for. The answer is C. I go through the question in detail in the full solution below (taken from the GMATFix App).

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by Brent@GMATPrepNow » Fri Mar 07, 2014 1:29 pm
LulaBrazilia wrote:If n = 4p, where p is a prime number greater than 2, how many different positive even divisors does n have, including n?

A) 2
B) 3
C) 4
D) 6
E) 8

Is it safe to plug in? Is there a quick algebraic approach?
Yes, plugging in a value works for this reason: let's say we choose some prime number value for p and then determine the number of different positive even divisors that n has. Since there can be ONLY ONE correct answer, choosing a different value should yield the same answer.

So, how about we let p = 3 (a prime greater than 2), which means n = 4(3) = 12
The even divisors of 12 are: 2, 4, 6 and 12
So, there are 4 even divisors.
Answer = C

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Brent
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by [email protected] » Fri Mar 07, 2014 3:06 pm
Hi LulaBrazilia,

TESTing Values works on lots of Quant questions on the GMAT. Here, you can use ANY prime number greater than 2 for P. If you're suspicious of your work, then you can always TEST two different values and watch what happens.

On this question, consider the following...

P=3; N = 12 and the positive EVEN divisors are 2,4,6 and 12 so the answer is FOUR
P=5; N = 20 and the positive EVEN divisors are 2,4,10 and 20 so the answer is also FOUR

Final Answer: C

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Rich
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