Problem Solving

And because of the rule jangojess cites above, perfect squares always have an *odd* number of divisors, because when you prime factorize a perfect square, the exponents are all even. When you add one to each power and multiply, you'll be multiplying odd numbers only, therefore getting an odd result. If you know that, you'll know without factorizing anything that the answer to the above question must be odd, and there's only one odd number among the answer choices.

Mozartain wrote:Hi Ian
could you please explain the rule Jangojess has given?

rather than explaining the rule with variables - which will get tedious, not to mention difficult to understand - i'll give an illustration with specific numbers.

let's say you want the total number of factors of the number 432, whose prime factorization is (2^4)(3^3).
here's the way to think about it: you have to choose (a) how many 2's, and, separately, (b) how many 3's are in the factorization. then you'll multiply these numbers of possibilities, in the same way as you'd multiply the numbers of possibilities for any other independent decision (such as 3 shirts x 4 pairs of pants = 12 outfits).

* because the factorization contains 2^4, there are 5 ways to choose the number of 2's: none, one, two, three, or four. note that this is 4 + 1 possibilities (= exponent + 1).
* because the factorization contains 3^3, there are 4 ways to choose the number of 3's: none, one, two, or three. note that this is 3 + 1 possibilities (= exponent + 1).
* then you'd multiply 5 x 4 = 20 possibilities total.

the same approach works in general, for any exponents that may happen to appear in a prime factorization. this is the genesis of the formula cited above.
of course, because TIME CONTRAINTS are a big deal on this exam, you should also just memorize the formula (i.e., you shouldn't have to derive it from scratch when you take the test, because that's a tremendous waste of time).

maxm wrote:The statement says: How many factors does 36^2 have?
So, it's asking for the total number of factors, aren't we me missing the negative facors?

So, it would be 25*2 = 50 factors total.
Isn't 25 the number of positive factors?

I'm kind of lost. Thanks for clarifying.

Good question. Normally, GMAT questions will restrict the values to consider only positive factors. In general, however, the GMAT allows for negative factors (divisors), as seen in this excerpt from the Official Guide:

An integer is any number in the set {. . . -3, -2, -1, 0, 1, 2, 3, . . .}. If x and y are integers and x " 0, then x is a divisor (factor) of y provided that y = xn for some integer n. In this case, y is also said to be divisible by x or to be a multiple of x. For example, 7 is a divisor or factor of 28 since 28 = (7)(4), but 8 is not a divisor of 28 since there is no integer n such that 28 = 8n.

Since the original question does not restrict the factors to only positive factors, the correct answer here is 50.

Cheers,
Brent