GMAT Number of Factors: How to Count the Possible Divisors of a Number

One of the recent additions to the GMAT content sphere is questions that ask you to count the total number of factors of a number rather than simply the number of prime factors.

That is, all factors, not just the prime ones. That’s any number that can divide evenly into the number you’re talking about. These questions might specifically ask you to count the number of factors, or they might ask you how many positive divisors a number has. These statements are equivalent.

(Counting the number of prime factors is easy: just do a prime factorization and count the number of bases.)

This is notably somewhat more difficult than counting simply the prime factors. In such a case, there is a simple procedure to use. It’s easy enough just to memorize, so here goes.

(Here I’m going to write GMAT number of factors for SEO.)

How to Count the Number of Factors for a Given Integer on the GMAT

First, prime factorize the number.

For example, if we wanted to know the number of factors for 36, we would prime factorize to get .

Second, add 1 to each of the powers:

Third, ignore the bases and multiply the powers together:

This will be the number of individual factors in the number 36. Don’t break your brain over this right now; the explanation will come later.

You can check this simply by counting out the factors for a number that’s fairly small, as 36 is. That is, the factors of 36 are: . As you can see, there are nine of them.

The first thing to do is to try this with a couple of different numbers.

84 has how many factors?

49 has how many factors?

144 has how many factors?

(Answers later)

So… the method is this:

Prime factorize your number

2) Add 1 to each power

3) Multiply these powers together: this is your total number of factors

Abstracted Questions about How to Count the Number of Factors on the GMAT

It’s also the reason that more advanced GMAT questions on this topic actually ignore the bases entirely: the number of possible factors actually has nothing to do with the bases at all. That is,

You might see a question phrased along the lines of: The number n can be written as the product of five prime numbers, two of which occur twice. How many different positive divisors does n have, including 1 and n?

In some sense, the explanation is almost easier using variables rather than actual numbers because we don’t have the confusion of the actual values of the bases to concern ourselves with (all the better, as these are irrelevant).

So what I would do in the above case is write out the strong of primes using variables. We know that there are five total numbers and two of them occur twice. That is:

Now, we know from above that the total number of factors will be to consider each of the powers plus one: .

Then we take the powers and multiply these: . That is, there should be 18 factors.

Here I am again writing GMAT number of factors. Ignore.

The Explanation — But Why?

Most explanations of this topic—if they address it at all—give it pretty short shrift. I’ve tried to give this a little bit more attention.

I would usually say that understanding the “why” behind something is massively useful on the GMAT, but I think this is a case where you can get by perfectly well without knowing precisely how the mechanism works. To be perfectly honest, I did this for years having simply memorized the procedure and not really digging into the “why” of it. No judgment.

But obviously there is some reason; otherwise we wouldn’t be able to get the answers to work out consistently.

That reason has to do with counting problems: combinatorics.

The number of possibilities of factors has to do with whether we are counting the individual number or not within our possibilities.

Let’s take a simple case of the number of possible factors from a number with two distinct prime factors. The prime factorization would then be .

Just by hand-counting, we know that our possibilities would be . That is, four factors.

We can universalize this by considering each of our primes, and , to be a slot.

__ * __

Now there are two possibilities for each slot. In the first slot, our possibilities are and 1. Basically, that’s whether we count or don’t count it; in the non-counting case, it becomes simply 1. I think of it as “turning the factor on or off,” where the on case counts and the off case counts 1.

So that means there are possibilities for : and 1. Same for . Therefore, the slots become this:

total factors.

In the case where we had, for example, appearing twice, we would no longer have 2 factors for the space. We would have three: . If appeared three times, we would have four factors: . And so forth.

Remember, the situation where the number is “turned off” means that 1 is a possibility for that slot. Essentially, the number of possibilities within a given slot will be the power of the number in the slot PLUS ONE to compensate for the “turned off” case.

So remember here that you’re not worried about the value of the prime number itself. It’s just how many different numbers can express within the individual slot. And that number will always be the power plus one.

Solutions

Oh no, it says “GMAT number of factors again.”

Checking the above, we see that the values will be as follows:

84 has how many factors?

The number 84 prime factorizes to . The number of possibilities for the first slot is , for the second is , and for the third is . The total number of possible factors here will be .

49 has how many factors?

The number 49 prime factorizes to . All we need to do is add 1 to the power here: is the total number of possible factors.
The number of possibilities is pretty obviously 3, even by hand-counting: .

144 has how many factors?

The number 144 prime factorizes to . The number of possibilities for the first slot is and the number of possibilities for the second slot is . Therefore, the total number of possible factors is .

And just for good measure, let’s write GMAT number of factors yet again because the algorithm is stupid and I hate it.