Factors
Factors
A factor is an integer that divides another integer evenly. If a/ b is an integer, then b is a factor of a. The numbers 3, 4, and 6, for example, are factors of 12.
Factorization
Sometimes the SAT requires you to find all the factors of some integer or to just be able to run through the factors quickly. To make this happen, write down all the factors of a number in pairs, beginning with 1 and the number you’re factoring. To factor 24:
  • 1 and 24 (1 × 24 = 24)
  • 2 and 12 (2 × 12 = 24)
  • 3 and 8 (3 × 8 = 24)
  • 4 and 6 (4 × 6 = 24)
If you find yourself beginning to repeat numbers, then the factorization’s complete. After finding that 4 is a factor of 24, the next lowest factor is 6, but you’ve already written 6 down. You’re done.
Prime Numbers
Everyone’s always insisting on how unique they are. Punks wear leather. Goths wear black. But prime numbers actually are unique. They are the only numbers whose sole factors are 1 and themselves. All prime numbers are positive (because every negative number has –1 as a factor in addition to 1 and itself). Furthermore, all prime numbers besides 2 are odd.
The first few primes, in increasing order, are
You don’t have to memorize this list, but getting familiar with it is a pretty good idea. Here’s a trick to determine if a number is prime. First, estimate the square root of the number. Then, check all the prime numbers that fall below your estimate to see if they are factors of the number. For example, to see if 91 is prime, you should estimate the square root of the number: . Now you should test 91 for divisibility by the prime numbers smaller than 10: 2, 3, 5, and 7.
  • Is 91 divisible by 2? No, it does not end with an even number.
  • Is 91 divisible by 3? No, 9 + 1 = 10, and 10 is not divisible by 3.
  • Is 91 divisible by 5? No, 91 does not end with 0 or 5.
  • Is 91 divisible by 7? Yes! 91 ÷ 7 = 13.
Therefore, 91 is not prime.
Prime Factorization
Come on, say it aloud with us: “Prime factorization.” Now imagine Arnold Schwarzenegger saying it. Then imagine if he knew how to do it. Holy Moly. He would probably be governor of the entire United States.
To find the prime factorization of a number, divide it and all its factors until every remaining integer is prime. The resulting group of prime numbers is the prime factorization of the original integer. Want to find the prime factorization of 36? We thought so:
It can be helpful to think of prime factorization in the form of a tree:
As you may already have noticed, there’s more than one way to find the prime factorization of a number. Instead of cutting 36 into 2 and 18, you could have factored it to 6 × 6, and then continued from there. As long as you don’t screw up the math, there’s no wrong path—you’ll always get the same result.
Greatest Common Factor
The greatest common factor (GCF) of two numbers is the largest factor that they have in common. Finding the GCF of two numbers is especially useful in certain applications, such as manipulating fractions (we explain why later in this chapter).
To find the GCF of two numbers, say, 18 and 24, first find their prime factorizations:
The GCF is the “overlap,” or intersection, of the two prime factorizations. In this case, both prime factorizations contain 2 × 3 = 6. This is their GCF.
Here’s another, more complicated, example: What’s the GCF of 96 and 144? First, find the prime factorizations:
96 = 2 × 48 = 2 × 2 × 24 = 2 × 2 × 2 × 12 = 2 × 2 × 2 × 2 × 6 = 2 × 2 × 2 × 2 × 2 × 3 = 25 × 3
144 = 2 × 72 = 2 × 2 × 36 = 2 × 2 × 2 × 18 = 2 × 2 × 2 × 2 × 9 = 2 × 2 × 2 × 2 × 3 × 3 = 24 × 32
The product of the “overlap” is 24 × 3 = 48. So that’s their GCF.
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