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Algebra I/Factoring/Factors of Integers

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Vocabulary

  • Factor: A factor of an integer n is any number that divides n without remainder.
  • Prime Number: A positive integer n is called prime if its only factors are 1 and n (and -1 and -n).
  • Composite Number: An integer n is called composite if it is not prime. This means that n is composite if it has a factor that is not 1 and not n.

Lesson

Sometimes numbers can be written as the product of other numbers. When we write a number n as a product n = a x b then say that a and b are factors of n. The equation n = a x b is called a factorization of n.

For example, 6 is a factor of 12 because 12 = 2 x 6. Also, 3 is a factor of 6 because 6 = 2 x 3. If we put these factorizations together we get 12 = 2 x 6 = 2 x (2x 3) = (2x2)x3 = 4 x 3 and so 3 and 4 are factors of 12.

If we have a number n we can always factor it as n = 1 x n. So 1 and n are always factors of n.

If we have a factorization n = a x b then n = (-a) x (-b). This means that if a is a factor of n then -a is also a factor of n. A positive integer can always be factored into positive integers.

We call a positive integer p a prime if it can only be factored into positive numbers as p = 1 x p or p = p x 1. The number 1 is a special number which we do not call prime.

There are many prime numbers: 2,3,5,7,11,13,17 and more.

When a positive integer is not prime we call it composite. Since we can write 12 = 2 x 6, we know that 12 is not prime. That means we call 12 composite.

There are many composite numbers: 4,6,8,12,14,15,16,18 and more.

Every positive number can be factored into a product of positive primes in only one way. For example, 30 = 2 x 15 = 2 x 3 x 5 where 2, 3 and 5 are prime. A factorization of a number into a product of primes is called a prime factorization. There is only one prime factorization of a number.

If we write a factorization of 30 then it must either contain a composite number or be 2x3x5. The factorizations of 30 are listed below:

  • 30 = 1 x 30,
  • 30 = 6 x 5,
  • 30 = 2 x 15,
  • 30 = 10 x 3, and
  • 30 = 2 x 3 x 5. This is the prime factorization of 30.

Example Problems

  1. Is 4 a factor of 20? Answer: Yes. The number 4 is a factor of 20 because 20 = 4 x 5.
  2. Is 6 a factor of 20? Answer: No. The number 6 is not a factor of 20 because 6 does not divide 20 without remainder; 20 = (6 x 3) + 2.
  3. Find all factorizations of 20. Which factorization of 20 is the prime factorization. Answer:
    • 20 = 1x20,
    • 20 = 2x10,
    • 20 = 4x5, and
    • 20 = 2x2x5 are the factorizations of 20. This last factorization is the prime factorization of 20.

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