How to Factor a Whole Number

Factoring of whole number denominators is a necessary step in finding a common denominator when adding and subtracting fractions. When factoring a quadratic equation, coefficients may also need to be factored. Furthermore, factoring a number into its prime factors seems like a simple enough task once it is demonstrated. The simplicity of it can, however, conceal the vital significance of the concept. Algebra students frequently have difficulties in working problems because they fail to see how to transfer the concept of factoring into the realm of variables. This step-by-step review will smooth the way for a student into algebraic factoring, which is critical to success in higher mathematics.

Instructions

1. Inspect the number to be factored and determine whether you recognize any numbers which will divide evenly into it. These divisors may, or may not, be primes. A prime number is one that can only be evenly divided by itself and one. The smallest primes include 2, 3, 5, 7, 11, 13, 17, 19 and 23. Remember that this article refers to factors that are not primes as divisors.
   The word "factor" will refer to prime factors.
   As a rule, the smaller the number, the easier it is to factor. Breaking it down into two smaller divisors will allow you to factor two smaller numbers that are much easier to factor. If you do not recognize any larger divisors, start with the smallest primes and try each one until you find one that will evenly divide into the number.
   Keep in mind the following facts:
   1) A number that ends in 0, 2, 4, 6 or 8 is even and can be evenly divided by 2.
   2) A number that ends in 0 or 5 can be evenly divided by 5.
   3) For all others, begin by trying the number 3. If 3 does not evenly divide into the number, try 7. Continue trying successively larger primes until you find one that divides evenly. When you have divided the original number, the result is called the quotient.
   Suppose that the number 252 is to be factored into its prime factors. Not seeing any larger divisors, and since the number is even, begin with the smallest prime which is 2; hence, 252 / 2 = 126

2. Repeat Step 1 using the indicated quotient (in this case, 126). The number 126 is also even, therefore it can also be divided evenly by 2; hence, 126 / 2 = 63

3. Continue to repeat Step 1 using each resulting quotient from the previous step.
   In this particular case, there are two possible courses. The number 63 is known to have divisors 9 and 7, but it is also divisible by 3, so proceed in the manner that is easiest for you.
   Dividing 63 by 3 leaves a quotient of 21, which is 3 times 7. On the other hand, breaking 63 up into its divisors, 9 and 7, leads to factoring 9 into 3 times 3. Either way, the result is the same.

4. Complete the factoring process by writing out each number that has been divided into the original number: 252 = 2 x 2 x 3 x 3 x 7.
   Once you have found all the prime factors of a number, all other divisors may be easily determined by using different combinations of these factors: 252 = 2 x 126 = 3 x 84 = 4 x 63 = 6 x 42 = 7 x 36 = 9 x 28 = 12 x 21 = 14 x 18.

Tips & Warnings

• Learning how to properly factor whole numbers is essential when learning fundamental and advanced algebra, and much higher math.