prime; factors

GMAT_crusher wrote:If the positive integer w has only two factors and one of these factors is v, where w and v are distinct values, which of the following must be true?

I. w > v
II. (5v)^2 > wv
III. w= (vw^3)/w^2

Help me out here guys .. I considered w=6; factors - 2 and 3. ... guide mentiones that w MUST be prime; factors - itself and 1.. hence v=1 !!!

How's that? .. In my case only I holds true, in prime case I and III hold true.

Please explain.

When GMAT questions mention factors, they refer to positive factors (most questions state so explicitly).

However, even without that explicit knowledge, if the integer ONLY has two factors, then it must be prime - the definition of a prime number is a number that has exactly two factors, 1 and itself.

You picked w=6, which doesn't follow the rule in the question, since 6 has 4 factors: 1, 2, 3 and 6. You can only consider numbers that follow all the rules.

Since w and v are distinct, and w has only two factors, v must be the other factor of the prime, which is 1.

So, we know that w can be any prime number and v must = 1.

Like Stuart said before:

1) If a number had only 2 factors IT MUST BE PRIME.

Let me add to this (since the GMAT does test this):

If a number has only 3 factors, then the number must be the SQUARE OF A PRIME.

For example:

Factors of 4 = 1, 2 , 4
Factors of 9 = 1, 3, 9

etc.

Thanks for the clarification guys.