Divisibility rules or Divisibility tests is the child techniques of Division operations in Mathematics.

Rule of Divisibility has developed to make division fast and better in worksheet, true or false, and with examples in Math subjects.

If you are interested to know about divisibility rule of 7, 11, 4 and 8, you must know division process deeply.

Because of, it is the parent of divisibility.

In mathematics Division is important part of four fundamental operations (Addition, Subtraction, & Multiplication are other three) which distributes any number into equivalent parts.

Division works just in opposite manner of multiplication and denoted by more than one symbol such as the slash (/), the horizontal line (|), and the division sign (÷).

## Divisibility Rules

The Divisibility rules (even without making large calculations) help to test whether a number can be completely divided by another number.

As mathematics student you must know about what are the divisibility rules.

Having knowledge about number divisibility rules for 2 to 20 is good for your study.

For example, divisibility rule of 9 will give the absolute answer whether the number is divisible by 9 or not.

It does not matter how big the number is. You can test the largest or the smallest number by this rule...

**Let’s dive and test few numbers below:**

### Divisibility rule of 7

The divisibility test of 7 is a complex task and requires some extra care to understand clearly.

Don’t worry we elaborate it in steps below.

- First take the last digit of the number, and double it. For instance in the number 343 last digit or unit digit is 3, so we have to double it means 3×2=6.
- Now subtract 6 (result of double last digit) from the rest number 34, therefore 34-6=28.
- Repeat the process until you get the smallest number.
- Now the remaining number is 28, And because 28 is divide by 7 exactly, hence the number 343 also divisible by 7

**Syntax: **

I) 905 (Double 5 is 10, 90−10=80, and 80÷7=11 3/7) Not Divisible by 7.

II) 672 (Double 2 is 4, 67−4=63, and 63÷7=9) Divisible.

### Divisibility rule of 11

What is the divisibility rule of 11? Answer given below:

Any number is divisible by 11 if the difference between the sum of digits at odd place and the sum of digits at even places is either 0 or a multiple of 11.

By following the rule, let’s check the number 2143, whether it is divisible by 11, or not?

- First split the number in group of alternative digit (odd place digit, and even place digit). And, because we're using 2143 in our example, so here we get 24 and 13 as alternative digit.
- Check the sum of each group i.e. 2+4=6 and 1+3= 4 in our example.
- Find the difference of the sum of the both i.e. 6-4=2.
- Check whether the result of the difference of the sum is divisible by 11 or not. If it's not, then the given number also not divisible by Eleven.
- Here the 2 is the difference which is not divisible by 11, so the given number 2143 also not divisible by Eleven. Example: let’s take 5123074 And 42031574.
- Sum of the odds places digits=(4+0+2+5)=11.
- Sum of the Even Places digits=(7+3+1)=11.
- Difference=11-11=0

Thus, 5123074 is divisible by 11, whereas in second case the difference is 14-12=2 As 2 is not divisible by 11, so the number 42031574 also not divisible by the eleven.

### Divisibility rule of 9

The rule for divisibility by 9 is similar to divisibility rule for 3. That is, if the sum of digits of the number is divisible by 9, then the number itself is divisible by 9.

**Let's Test 902358 & 123540 **

- Sum of the digits=9+0+2+3+5+8=27.
- As the result i.e. 27 is divisible by 9, So 902358 also divisible by nine.
- Sum of the digits: 1+2+3+5+4+0=15, So 15 As well as number itself not divisible by 9.

### Divisibility rule of 8

If the last three digits (of course on Hundreds, tens and one's place Digit) of a number are divisible by 8, then the number is completely divisible by 8.

Test: 45064 (064÷8=8) Yes, 77583(583÷8=72.875) No

### Divisibility rule of 4

Given number is divisible by 4 if the last two digits (tens and unit place) of the number are divisible by 4 exactly.

The same rule also applied for check divisibility of 25.

Let’s test 3016, 74048,502352 and 6320456 All are divisible by 4 as 16, 48, 52 and 56 are divisible by 4 without reminder.

While the numbers 1253, 74327, 66341 and 420035 not divisible, because of tens and unit place digits are 53, 27, 41 and 35 that not divide by 4 exactly.

### Divisibility rule of 3

Rule of divisibility for 3 is similar to 9. The rule states that the number is completely divisible by 3 if the sum of its digits is divisible by 3. e.g. 135603 Sum of digits=1+3+5+6+0+3=18

Test: 18/3=6. As 18 divisible by 3, so the number is also divisible.

### Divisibility Rule of 6

- Numbers which are divisible by both 2 and 3, also divisible by 6.
- If the last digit of the given number is even and the sum of its digits is a multiple of 3, then the given number is also a multiple of 6.

**Test 1****: Check 230412, 413037, 7103284**

230412, the number satisfies the rule as number is divisible by 2 as the last digit is 2.

Sum of Digits=2+3+0+4+1+2=12, as the result 12 is divisible by both 2 & 3,so number also divisible by 3

**Test 2:**

413037, as the last digit is 7, which not fits in rule exactly divide by 2.

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