If P and Q are positive integers, what is the number of factors of X?

Question

1) X = P * Q; P and Q have exactly one common prime factor.

2) P and Q each have exactly three factors.

Source: Expert Global

IanStewart · Answer

Neither statement is close to being sufficient alone.

The only numbers that have exactly three divisors are squares of primes, so numbers with prime factorizations that look like p^2. That number has three divisors: 1, p and p^2. So if Statement 2 is true, P = p^2 and Q = q^2, where p and q are prime, and if Statement 1 is true, and these numbers share a prime divisor, it must be true that p = q, and both numbers are equal to p^2. So from Statement 1, x = PQ = (p^2)(p^2) = p^4, which has five divisors, and the answer is C.

Mo2men · Answer

Hi  IanStewart

Thanks for your explanation.

I have a quick question regarding statement 1. It says that both share one common prime factor. Is the following example would satisfy statement 1?

P= 2^2 * 5 & Q= 2^2 * 5

Mu understanding is that both share unique prime number 2 but both share 2^2 which two prime numbers (2 & 2). I'm confused to apply statement 1 in case of that example given.

thanks

IanStewart · Answer

In your example, P and Q share two prime factors: 2 and 5. Perhaps you meant to give an example more like this:

P = (2^4)(5)
Q = (2^9)(11)

In this case, P and Q share only one prime factor, 2. There is no reason to count the '2' more than once; if you did, you'd be answering a different question.

Mo2men · Answer

Thanks  IanStewart

You are correct about understanding me. I meant something like your example.

But what do you mean by "if you did, you'd be answering a different question."? When I should count the prime numbers? As far as I know, there isa difference between number of prime factors & number of unique prime factors?

IanStewart · Answer

If you asked a mathematician "how many prime factors does (2^6)(5^8) have?", they would say "two", because there are two prime numbers that divide (2^6)(5^8), namely 2 and 5. I'm not exactly sure how someone would justify counting the 2 six times, and the 5 eight times. Now, it is true in actual Number Theory that you sometimes care about counting repeated prime divisors, when you are finding something called the "length" of a number (there is one old GMAT question that tests this, but it tells you the definition of "length" in the question itself). For that reason, the GMAT will normally use the phrase "distinct prime divisors" to avoid any potential confusion, but the word "distinct" isn't really necessary.

You can see why the interpretation I'm making is the logically correct one by asking a different question: "how many divisors does 256 have?" You wouldn't count "4" more than once, even though you can divide 256 by 4 a few times. So if I ask "how many prime divisors does 256 have?" you similarly would not count '2' more than once, even though you can divide 256 by 2 several times.

Mo2men · Answer

Thanks a lot  IanStewart  for your care and patience to reply and it all makes sense too.

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If P and Q are positive integers, what is the number of factors of X?

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