Basic Algebra/Factoring/Factors of Integers
- 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.
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.
But, for example, 4 is not a factor of 13 because 4 cannot multiply any other whole number and come out with an even 13.
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 that then it must either contain a composite number of 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.
What, then, is the prime factorization of 100?
100 = 1 x 100, 100 = 2 x 50 100 = 4 x 25 100 = 2 x 2 x 5 x 5
- Is 4 a factor of 20? Answer: Yes. The number 4 is a factor of 20 because 20 = 4 x 5.
- 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.
- 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.
One way to think of factor games is as all the different ways you can package groups of given numbers. For instance 12 items can be packaged as a string of 12, like a giant skinny chocolate bar, or as 2 groups of six like a carton of eggs, or as 4 sets of 3 like two six packs of soft drinks. Experience with factors and numbers can make other people respect your ideas when you use your number sense to find more attractive or efficient ways to create or display something.
There are many online games you can play involving factorization. One of my favorites is the factor game. In order to win this game you must pick numbers with small prime factors while leaving numbers with special large composite factors for the end game.
A factor game you can play without a computer is to pick a range of prime numbers. You can give one player a time limit to construct a composite number with these primes. If there are just two players you can allow player number 2 to factor the number and then compare the factorizations. If player 1 made a mistake constructing the number then player 2 gets a point, if player 2 does not factor the number correctly then player 1 gets a point. This game can be extended for multiple players by allowing the players factoring the numbers to race for the factorization, and then checking the answers in the order which they are turned in. Something that makes this game interesting is that multiplications with larger prime numbers become harder to perform correctly and harder to decompose. You can easily handicap this game by giving some players more times to pick their number then other players. For instance how big a number can you build with the primes 2, 3, 5, 7, 11, and 13 in 30 seconds? It takes less time to double numbers than multiply them by 11, but these numbers are also easier to factor.