This article describes a simple method that can be used to factorize any integer into its prime factors. Although the vast majority of school math work deals with smaller numbers, large numbers may be factorized using this method too.
Prime Numbers Up To 50
The first step in calculating the prime factors of a number is to know the prime numbers. The first ten prime numbers are:
2
3
5
7
11
13
17
19
23
29.
In almost all basic grade-school or high school arithmetic, it is seldom necessary to use prime factors higher than 7. It is possible to check whether a number is divisible by the numbers from 2 to 11, without having to actually do the division, and this can save time in the factorization process.
Prime Factorization Method: Example 504
The steps to finding the prime factors of a number (example 504) are:
1. Starting with the lowest prime number 2, check if it divides the number 504 without a remainder and go to step 2. If not, go to step 4.
2. Divide 504 by 2 to give 252.
3. Repeat this division by 2 until a number is reached that is not divisible by 2:
504÷2 = 252
252÷2 = 126
126÷2 = 63
63÷2 leaves a remainder, so go to step 4.
4. Starting with the last number from step 3, repeat the steps 1 to 3 for the next highest prime number (in this case 3).
63 is divisible by 3, so
63÷3 = 21
21÷3 = 7
7÷3 leaves a remainder, so go to next step
5. The next prime number up from 3 is 5. Since 7 is not divisible by 5, go to the next highest prime number on the list, which is 7.
6. 504 has been divided down to 7, and this is then divided by the next highest prime number on the list, 7:
7÷7 = 1
7. When the original number has been divided down to 1, the process stops. If the final number cannot be reduced to 1, then the final number is a prime number, and should be listed as a factor of the original number.
Listing The Prime Factors
The prime factors of 504 have now been found, and they need to be "summarized". This is to make it easy to manipulate the prime factors, especially when comparing pairs or sets of numbers with a view to finding the lowest common multiple or greatest common divisor.
504 was divided by 2 a total of three times to give 63. This is the same as dividing by 8 just once, and it is commonly written as 2 x 2 x 2.
63 was divided by 3 twice, or 3 x 3, to give 7.
7 was divided by 7 once, to give 1.
So the factors of 504 are 2 x 2 x 2 x 3 x 3 x 7. This is quite long to write, so it is usually written as 2³ x 3² x 7, or 2^3 x 3^2 x 7.
Prime Factorization Summary
The three simple steps to factorize a number into it's prime factors are:
List the prime numbers
Divide the number by each prime number, as many times as can be done without leaving a remainder.
Continue until the number is eventually divided down to 1.
List the prime factors in shorthand form for convenience.
Prime Factors References
The book " Tables of the Prime Numbers, and Prime Factors of the Composite Numbers, From 1 to 100,000: With the Methods of Their Construction, and Examples of Their Use " contains a list of the prime numbers, as well as a description of the method used here.
Martin Bell - Martin holds a B.Sc. degree in chemical engineering, and an M.Sc. degree in electronics and computing. He has spent more than 25 years ...
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