Factor. Factors of a number are numbers which, when multiplied together, give the original number. For example, the numbers 3 and 4 are factors of 12 because 3 X 4 = 12. The other whole number factors of 12 are 2 and 6, and 1 and 12. Factoring (determining factors) provides insight into one of the many relationships among numbers.
Every whole number, except 1, can be expressed as the product of at least two factors. A number that has exactly two different factors, itself and 1, is called a prime number. The number 7 is prime because 1 and 7 are its only factors. The eight smallest primes are 2, 3, 5, 7, 11, 13, 17, and 19. A number that has more than two factors is called a composite number. The number 4 is composite because it has three factors, 1, 2, and 4. The eight smallest composite numbers are 4, 6, 8, 9, 10, 12, 14, and 15. The number 1 is neither composite nor prime.
Prime factors
of a number are those prime numbers which, when multiplied together, equal the number. Each composite number is a product of only one set of prime numbers. For example, 24 can only be expressed as a product of prime numbers as 2 X 2 X 2 X 3 (in any order). The prime factorization of 24 is 2 X 2 X 2 X 3, and the prime factors of 24 are 2 and 3.
To find the prime factors of a number, divide the number by any prime number that divides it evenly. It is usually easiest to use the smallest prime number that divides the number evenly. For example, to find the prime factors of 220, begin by dividing by 2 (220 divided by 2 = 110). Continue dividing the quotient (the number obtained) by 2 until it is no longer divisible by 2 (110 divided by 2 = 55). But 55 cannot be divided by 2 without leaving a remainder. The next prime, 3, does not divide 55 without a remainder either. But the next greater prime, 5, does divide 55 equally (55 divided by 5 = 11). The number 11, like 2 and 5, is a prime number. Therefore the prime factorization of 220 is 2 X 2 X 5 X 11, and the prime factors are 2, 5, and 11. The product 2 X 2 X 5 X 11 (in any order) is the only way 220 can be expressed as the product of prime numbers.
Common factors.
If a number is a factor of two or more numbers, it is called a common factor of those numbers. For example, 1, 3, 5, and 15 are the factors of 15; and 1, 2, 4, 5, 10, and 20 are the factors of 20. The numbers 1 and 5 are common to both these sets of factors. The numbers 30 and 45 have four common factors: 1, 3, 5, and 15.
If two numbers have more than one common factor, the greatest one is called the greatest common factor. It is also the greatest common divisor because a factor of a number is also a divisor of that number. To find the greatest common factor of two or more numbers, first find the set of all the factors for each number. Then select the largest factor that is in all the sets. For example, here are the factors for 18 (1, 2, 3, 6, 9, 18); 30 (1, 2, 3, 5, 6, 10, 15, 30); and 42 (1, 2, 3, 6, 7, 14, 21, 42). The number 6 is the greatest factor common to all the sets, so 6 is the greatest common factor of 18, 30, and 42.
Relative primes.
Two numbers that have no common factors other than 1 are relatively prime or prime in relation to each other. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12. The factors of 35 are 1, 5, 7, and 35. Twelve and 35 have no common factors other than 1. They are relatively prime.
Algebraic factors.
Algebraic expressions, which use letters to represent unknown numbers, also have factors. The factors of 3ab, for example, are 1, 3, a, b, 3a, 3b, ab, and 3ab.
Factoring can help simplify more complicated algebraic expressions, such as 5x + 5. This expression is a product of two factors: 5 and (x +1). Such an expression might be easier to work with when written as 5(x+ 1) instead of 5x + 5. Likewise, the expression 2a 2 + 4ab can be written as the product of two of its factors: 2a and (a + 2b).
See also Algebra; Multiplication; Number theory; Prime number; Sieve of Eratosthenes.
