If n is not a prime number, then is the product of all the positive fa

Question

If n is not a prime number, then is the product of all the positive factors of n a perfect square?

(1) n is the smallest positive integer which is not a factor of 63!

(2) n and 63! has exactly one common prime factor.

SHoylandBPrep · Answer

The positive factors of a number usually come in pairs. For instance, the factors of 15 are 1 × 15 and 3 × 5. The product of each of these pairs is n, so when you multiply all these pairs together, you get a perfect square if there is an even number of pairs. For instance, n = 15 has two pairs, so the product of its factors is a perfect square. But n = 20 has three pairs, so the product of its factors is not a perfect square.

Stat. (1) 
There is a one specific value that n has to be -- it happens to be n = 67. Therefore, we have a definite answer to the question.

Sufficient

Stat. (2) 
You can use a multiple of 67 for n, like n = 134 or n = 201, which have an even number of pairs.

To get an odd number of pairs, you could use a number like n = 2 × 67² has an odd number of pairs:

→ 1 and 2 × 67²

→ 2 and 67²

→ 67 and 2 × 67

The product of these factor is not a perfect square because there are three 2s altogether.

Insufficient

(a) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.

IanStewart · Answer

The question tells us in the stem that "n is not a prime number", but then Statement 1 tells us n is 67, which is a prime number. So the question makes no logical sense.

Regardless, any statement similar to Statement 1 must be sufficient, because Statement 1 tells us exactly what n is. If we can find the number n, we can answer any question about n. There's no reason to even think about what the answer to the question is.

Using Statement 2 (if I understand it correctly - I don't know why the verb 'has' is singular), n might equal 4 = 2^2, and the product of 4's factors is not a square (it is 8), or n might equal 8 = 2^3, and the product of 8's factors is a square (it's 64), so Statement 2 is not sufficient.

SohGMAT2020 · Answer

I believe for statement 1 n will be the smallest multiple of 67 ie 67 * 2 = 134 and the product of factors will be ( 2 * 67 and 134 * 1 ) - 134^2 is a perfect square - sufficient ( Since 134 is not a factor of 63! and not prime as well)

IanStewart · Answer

It's very possible that's what the question intends, but it's certainly not what the question actually says.

In any case, if Statement 1 defines a unique numerical value of n in some way, it's automatically sufficient if we're asked a question about n -- there's no need to do any work at all when that's true.

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If n is not a prime number, then is the product of all the positive fa

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If n is not a prime number, then is the product of all the positive fa

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