Data Sufficiency

Gmat_mission wrote:If x is a prime number, what is the number of factors of 75x?

1) x^2 has 3 factors
2) x>5

[spoiler]OA=B[/spoiler].

I don't understand how should I solve this DS question? May any expert give me some help? Thanks in advanced.

Given: x is a prime number

We have to find out the number of factors of 75x.

First, factorize 75x.

75x = 3.5^2.x

Case 1: Say x is a prime number other than 3 and 5; say x =7

=> 75x = 3.5^2.7 = 3^1.5^2.7^1 = = 3^a.5^b.7^c, where a = 1, b = 2 and c =1

The number of factors of 75x = (a+1)*(b+1)*(c+1) = (1+1)*(2+1)*(1+1) = 2*3*2 = 12

Case 2: Say x is a prime number between 3 and 5; say x =3

=> 75x = 3.5^2.3 = 3^2.5^2 = = 3^a.5^b, where a = 2, b = 2

The number of factors of 75x = (a+1)*(b+1) = (2+1)*(2+1) = 3*3 = 9

Case 3: Say x is a prime number between 3 and 5; say x =5

=> 75x = 3.5^2.5 = 3^1.5^3 = = 3^a.5^b, where a = 1, b = 3

The number of factors of 75x = (a+1)*(b+1) = (1+1)*(3+1) = 2*4 = 8

Since the number of factors of 75x is not equal, we must identify whether x is 3, 5 or other than them.

Let's take each statement one by one.

1) x^2 has 3 factors.

=> x can be any prime number: 3, 5 or other. Insufficient.

If x = 3, then the number of factors of 3^2 = 9 are 1, 3, and 9. The same goes for any prime number.

2) x > 5

=> x is a prime number other than 5. Sufficient. The number of factors of 75x is 12 (Case 1). Sufficient

The correct answer: B

Hope this helps!

-Jay
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