[MathI] 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.
    Sometimes it is necessary or helpful tofactor an integer completely. This means you need to find all thefactors of that integer. It’s possible that the test will directlyrequire this skill or will make use of it in a more complicatedquestion. In either case, it’s something you should know how to do.

    Factorization

    To find all the factors of a number, writethem down in pairs, beginning with 1 and the number you’re factoring.We’ll factor 24 in this example. One and 24 are both factors of 24.Next, try every integer greater than 1 in increasing order. Here arethe factor pairs we find for 24:

• 1 and 24 (124 = 24)
• 2 and 12 (212 = 24)
• 3 and 8 (38 = 24)
• 4 and 6 (46 = 24)

    You know you’ve found all the factors of anumber when the next first factor exceeds its corresponding secondfactor. For example, after you found that 4 was a factor of 24 and 5was not, you would see that 6, the next factor of 24, had already beenincluded in a pair of factors. Thus, all the factors have been found.

    Prime Numbers

    A prime number is a number whose only factors are 1 and itself. 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:

    To determine whether a number is prime, youshouldn’t check whether the number is divisible by every number lessthan itself. Such an effort would take an incredible amount of time,and you have only an hour for the Math IC. Instead, to decide whether anumber is prime, all you need to do is estimate the square root of thenumber, then check all the prime numbers that fall below your estimate.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! 917 = 13.

    Therefore, 91 is not prime.

    Prime Factorization

    Another form of factorization is called prime factorization. The prime factorization of an integer is the listing of the prime numbers whose product is that number.

    To find the prime factorization of a number,divide it and all its factors until every remaining integer is prime.This group of prime numbers is the prime factorization of the originalinteger. As an example, let’s find the prime factorization of 36.

    It can be helpful to think of prime factorization in the form of a tree:

    As you may already have noticed, there ismore than one way to find the prime factorization of a number. We couldhave first resolved 36 into 6 6, for example, and then determined the prime factorization from there.So don’t worry—you can’t screw up. No matter which path you take, youwill always get the same result. That is, as long as you do yourarithmetic correctly. Just for practice, find the prime factorizationsfor 45 and 41.

    Since the only factors of 41 are 1 and 41, 41 is a prime number. It is therefore its own prime factorization.

    Greatest Common Factor

    The greatest common factor (GCF) oftwo numbers is the greatest factor that they have in common. Findingthe GCF of two numbers is especially useful in certain applications,such as manipulating fractions (we explain why later in this section).

    In order to find the GCF of two numbers, wemust first produce their prime factorizations. What is the greatestcommon factor of 18 and 24, for example?

    First, their prime factorizations:

    The greatest common factor is the greatestinteger that can be written as a product of common prime factors. Thatis to say, the GCF is the “overlap,” or intersection, of the two primefactorizations. In this case, both prime factorizations contain 2 3 = 6. This is their GCF.

    Here’s another example:
What is the GCF of 96 and 144?

    First:

    So, the product of the prime factors that they share is 24 3 = 48, which is their GCF.

    For practice, find the GCF of the following pairs of integers:

• 12 and 15
• 30 and 45
• 13 and 72
• 14 and 49
• 100 and 80

    Compare your answers to the solutions:

• 12 = 22 3. 15 = 3 5. The GCF is 3.
• 30 = 2 3 5. 45 = 32 5. The GCF is 3 5 = 15.
• 13 = 1 13. 72 = 23 3. There are no common prime factors. The GCF is 1.
• 14 = 2 7. 49 = 72. The GCF is 7.
• 100 = 22 52. 80 = 24 5. The GCF is 22 5 = 20.

    Relatively Prime Numbers

    Two numbers are called relatively primeif they have no common prime factors (i.e., if their GCF is 1). Thisdoesn’t mean, however, that each number is itself prime. The numbers 8and 15 are relatively prime because they have no common primes in theirprime factorizations (8 = 2 2 2 and 15 = 3 5), but neither number is prime.