In the Name of Allah; Most-Merciful, Most-Compassionate.
Divisibility rules:
Divisibility rules of whole numbers are very useful because they help us to quickly
determine if a number can be divided by 2, 3,4,5,6,7,8,9, and 10 without doing long
division.
Divisibility means that you are able to divide a number evenly.
For instance, 9 can be divided evenly by 3 because 9/3 = 3. However, 8 cannot be
divided evenly by 3.
How can we tell if a number can be divided exactly by by 2 ;3;4;5;6;7;8;9 and 10 ?
Dividing by 2
All even numbers are divisible by 2. Which means that all numbers ending in 0,
2,4,6 or 8 are divided by 2
Example:
5487925836 and 254879638 are divided by 2
748965107 and 54824763 are not divided by 2
Dividing by 3
Add up the digits of the number, if the sum is divisible by three, then the number is as
well.
Example1:
111111
1. Add up all the digits in the number. the digits add to 6; (1+1+1+1+1+1=6)
2. the sum is 6. If the sum is divisible by 3, so is the number.
Example2:
12123
(1+2+1+2+3=9) 9 is divisible by 3, therefore 12123 is too
Dividing by 4
1. Are the last two digits in your number divisible by 4?
2. If so, the number is too!
Example: 358912
ends in 12 which is divisible by 4, thus so is 358912.
Dividing by 5
1. Numbers ending in a 5 or a 0 are always divisible by 5.
Example: 358912 is not divisible by 5
254870 is divisible by 5
Dividing by 6
1. If the Number is divisible by 2 and 3 it is divisible by 6 also.
Example:
102 is divisible by 2 and 3, thus divisible by 6
Dividing by 7 (two Tests)
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FIRST TEST
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Take the last digit in a number.
Double and subtract the last digit in your number from the rest of the digits.
Repeat the process for larger numbers.
Example1: 364 (the last digit is 4; Double the 4 to get 8. Subtract 8 from 36 to get
28 which is divisible by 7 and we can now say that 364 is divisible by 7.
Example2: 224
Double the 4 to get 8. Subtract 8 from 22 to get 14 which is divisible by 7
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NEXT TEST
Take the number and multiply each digit beginning on the right hand side (ones)
by 1, 3, 2, 6, 4, 5. Repeat this sequence as necessary
Add the products.
If the sum is divisible by 7 - so is your number.
Example: Is 2016 divisible by 7?
6(1) + 1(3) + 0(2) + 2(6) = 21
21 is divisible by 7 and we can now say that 2016 is also divisible by 7.
Is 348 divisible by 7?
Remove the last digit, which is 8. The number becomes 34. Then, Double 8 to get 16
and subtract 16 from 34.
34 − 16 = 18 and 18 is not divisible by 7. Therefore, 348 is not divisible by 7
Is 37961 divisible by 7?
Remove the last digit, which is 1. The number becomes 3796. Then, Double 1 to get
2 and subtract 2 from 3796.
3796 − 2 = 3794, so still too big? Thus repeat the process.
Remove the last digit, which is 4. The number becomes 379. Then, Double 4 to get 8
and subtract 8 from 379.
379 − 8 = 371, so still too big? Thus repeat the process.
Remove the last digit, which is 1. The number becomes 37. Then, Double 1 to get 2
and subtract 2 from 37.
37 − 2 = 35 and 35 is divisible by 7. Therefore, 37961 is divisible by 7.
For instance, 587320 is divisible by 8 because 320 is divisible by 8.
Dividing by 8
1. If the last 3 digits are divisible by 8, so is the entire number.
Example: 6008 - The last 3 digits are divisible by 8, therefore, so is 6008.
587320 is divisible by 8 because 320 is divisible by 8.
Dividing by 9
1. Almost the same rule and dividing by 3. Add up all the digits in the number.
2. Find out what the sum is. If the sum is divisible by 9, so is the number.
For example: 43785 (4+3+7+8+5=27) 27 is divisible by 9, therefore 43785 is too!
Dividing by 10
1. If the number ends in a 0, it is divisible by 10