LESSON: Prime Factorization

READ: Use Divisibility Rules to Find Factors

Use Divisibility Rules to Find Factors of Given Numbers

When we have a larger number that we are factoring, we need to use divisibility rules to help us find the factors of that number.

What are divisibility rules?

Divisibility rules looks at different numbers in a given number. Depending on the numbers or combinations of numbers, we can determine if a number is divisible by let’s say 2 or 3 or 4. This can help us to identify the factors of a number.

Here is a chart that shows all of the divisibility rules.

Characteristic of Number	Number It’s Divisible By	Example:
Last digit is even	2	208
The sum of all the digits is divisible by 3	3	513 5 + 1 + 3 = 9 9 is divisible by 3, so 513 is also divisible by 3.
The last two digits are divisible by 4	4	616 16 is divisible by 4, so 616 is also divisible by 4.
The last digit is 0 or 5	5	590
The number is divisible by 2 and 3	6	438 Last digit is even and 4 + 3 + 8 = 15, which is divisible by 3.
Double the last digit, subtract it from the rest of the number (not including the last digit). If that number is divisible by 7, the original number is divisible by 7.	7	574 4×2 = 8 57 – 8 = 49 49 is divisible by 7, so 584 is also divisible by 7.
The last 3 digits are divisible by 8	8	1,856 856 is divisible by 8, so 1,856 is also divisible by 8.
The sum of the digits is divisible by 9	9	567 5 + 6 + 7 = 18 18 is divisible by 9, so 567 is also divisible by 9.
The last digit is a 0	10	1,560
The number is divisible by both 3 and 4	12	1,824 1 + 8 + 2 + 4 = 15, which is divisible by 3. 24 is divisible by 4, so 1,824 is also divisible by 4. Since 1,824 is divisible by 3 and 4, it is also divisible by 12.

Now some of these rules are going to be more useful than others. But you can use this chart to help you.

Example

What numbers is 1346 divisible by?

To solve this, we can go through each rule and see if it applies.

1. The last number is even-this number is divisible by 2.
2. The sum of the digits is 14-this number is not divisible by 3.
3. The last two digits are not divisible by 4-this number is not divisible by 4.
4. The last digit is not a zero or five-this number is not divisible by 5.
5. 6x2=12; 134 - 12 = 122. 122 is not divisible by 7-this number is not divisible by 7.
6. 346 is not divisible by 8-this number is not divisible by 8.
   
7. The sum of the digits is 14-this number is not divisible by 9
8. The number does not end in zero-this number is not divisible by 10
9. The number is not divisible by 3 and 4.

Our answer is that this number is divisible by 2.

You won’t usually have to go through each rule of disability, but it is important that you know and understand them just in case.