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Reducing Numbers to Prime Factors
by Ron Kurtus (11 January 2008)
Every number is either a prime number or is a multiple of two or more primes. A prime number is only divisible by itself and 1. You can reduce a number into its prime number factors by dividing the number by the various primes. Much of this is by trial and error, but there are some tricks to in finding certain prime factors. Reducing the number into its prime factors is important in being able to reduce fractions to their lowest possible terms.
Questions you may have include:
- What are the prime numbers?
- What are tricks to use to find prime factors?
- What are some examples of reducing a number to its prime factors?
This lesson will answer those questions. There is a mini-quiz near the end of the lesson.
Prime numbers
A prime number is a number that can only be divided by itself and 1. The first ten prime numbers are:
2
3
5
7
11
13
19
23
29
31
There is no formula for finding prime numbers, but you should at least know the first several of them.
Tricks in finding prime factors
You typically start to reduce a number to its factors with the smallest primes, and then you work your way up. there are some tricks to remember to help you quickly tell if the number is divisible by the certain prime.
Divisible by 2
The number 2 is the smallest prime. You can tell if a number is divisible by 2 if the last digit is an even number. Examples are 6, 18, 250, 1354432 and so on.
Divisible by 3
A trick to see if a number can be divided by 3 is to add the digits together to see if they can be divided by 3. For example 12 is divisible by 3 since 1 + 2 = 3. Likewise, 51 is divisible by 3 since 5 + 1 = 6. Also, 765 is divisible by 3 since 7 + 6 + 5 = 18 and 1 + 8 = 9.
Divisible by 5
Numbers divisible by 5 end in either 5 or 0.
Other primes
Unfortunately, I don't know of any other simple tricks that find out if numbers are divisible by other primes.
Reduce to prime numbers
You can break a number to its factors by trial and error. Take the number and see if there are any tricks that apply. Then try to divide the number by various primes, starting with the smallest likely prime. If you can't find any primes that work, then the number may also be a prime.
Continue the process until you have the number broken into a series of primes.
Example: number 12
Consider the number 12.
You know it is divisible by 2, which is a prime. 12 ÷ 2 = 6.
But 6 is also divisible by 2, such that 6 ÷ 2 = 3. And 3 is a prime number.
Thus, 12 = 2 × 2 × 3.
Example: number 147
Consider 147.
You can see it is not an even number.
Is it divisible by 3? A trick to see if a number can be divided by 3 is to add the digits together to see if they can be divided by 3: 1 + 4 + 7 = 12 and 1 + 2 = 3. Thus 147 is divisible by 3. So, 147 = 3 × 49.
Now 49 is not divisible by 3 since 4 + 9 = 13. The next prime is 5 and 49 obviously is not divisible by 5. Only numbers ending in 5 or in 0 are divisible by 5. The next prime is 7, and 49 ÷ 7 = 7.
Thus 147 = 3 × 7 × 7.
Example: number 149
Consider 149.
It is odd, so it is not divisible by 2.
Since 1 + 4 + 9 = 14, it is not divisible be 3.
It doesn't end in 0 or 5, so it is not divisible by 5.
149 ÷ 7 does not work.
Using the trusty calculator, note that 149 ÷ 11 = 13.54. It is not divisible by 11, and thus is is not divisible by 13 either. Dividing by larger primes will not result in smaller primes, since we already tried them. Thus, we can conclude the 149 is a prime number.
Probably won't have to do that
Now you probably (hopefully) will never have to determine if a number that size or larger is a prime, but at least you know how to make that determination.
Summary
Every number is either a prime number or is a multiple of two or more primes. A prime number is only divisible by itself and 1. You can reduce a number into its prime number factors by dividing the number by the various primes. Much of this is by trial and error, but there are some tricks to in finding certain prime factors.
Be a prime candidate for success
Resources
The following resources provide information on this subject:
Websites
Books
Mini-quiz to check your understanding
1. What is the eleventh prime number?
2. What are the prime factors of 144?
3. What are the prime factors of 143?
If you got all three correct, you are on your way to becoming a Champion in Arithmetic. If you had problems, you had better look over the material again.
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Reducing Numbers to Prime Factors