Numbers With Three Factors

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Factors and multiples appear on a large number of GMAT math problems. One subset of those questions concerns how many factors a given number has, which I covered in another article.

More interesting are properties of classes of numbers, and how many factors they have. For instance, prime numbers have only two factors: one and itself. It's also worth understanding that the only numbers with two factors are prime.

How about three factors? What are the properties of numbers that have as factors one, itself, and one other number?

As it turns out, the only positive integers with exactly three factors are the squares of primes. For instance, the factors of 9 are 1, 3, and 9, and the factors of 49 are 1, 7, and 49.

Odd numbers of factors

If you find all of the factors of a non-square, you can "pair off" the factors. For instance, the factors of 12 are 1, 2, 3, 4, 6, and 12. You can split those six factors into three pairs, each of which multiplies to 12:

  • 1 and 12
  • 2 and 6
  • 3 and 4

However, if you try the same thing with a square, you end up with a duplicate. The factors of 16 are 1, 2, 4, 8, and 16, some of which pair off:

  • 1 and 16
  • 2 and 8
  • 4 and ... itself

Any time you are finding the factors of a square, the final step will involve the square root, like 4 above. That square root only counts once--4 is only one factor, not two. So 16, like 9 and 49, has an odd number of factors.

We can generalize this and state the rule in a couple of different ways. First, if an integer has an odd number of factors, it is a perfect square. Second, if a number is a perfect square, it has an odd number of factors.

Most of the time this comes up on GMAT questions, you'll encounter the concept of integers with three factors. However, it's a small step from there to understanding the more universal concept, which is likely to appear on much more challenging test items.



About the author: Jeff Sackmann has written many GMAT preparation books, including the popular Total GMAT Math, Total GMAT Verbal, and GMAT 111. He has also created explanations for problems in The Official Guide, as well as 1,800 practice GMAT math questions.

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