Mathematics is not very easy for some of us. At times, the need for tricks and shorthand techniques is felt, so as to solve math problems faster and easier without actual calculation. Divisibility rules are a great example of such shorthand techniques.

As its name suggests, divisibility rules help one to check whether a number is divisible by another number without actual division. If a number is completely divisible by another number it means that in such a case quotient will be a whole number and the division will leave zero as remainder. Since every number is not completely divisible by every other number such numbers leave remainder other than zero. Divisibility rules are certain rules which help us to determine the actual divisor of a number just by considering the digits of the number. Let us look into these rules for different whole numbers one by one.

**Divisibility by 1: **Every number is divisible by 1. Divisibility rule for 1 doesn’t have any particular condition. Any number divided by 1 will give the number itself, irrespective of how large the number is. For example, 3 is divisible by 1 and 3000 is also divisible by 1 completely.

**Divisibility by 2: **Any even number or number whose last digit is an even number i.e. 2,4,6,8 including 0 is always completely divisible by 2.

**Example: **508 is an even number and divisible by 2 but 509 is not an even number, hence not divisible by 2. Procedure to check whether 508 is divisible by 2 or not is as follow:

- Consider the number 508
- Just take the last digit 8 and divide it by 2
- If the last digit 8 is divisible by 2 then the number 508 is also divisible by 2.

**Divisibility by 3: **Divisibility rule for 3 states that a number is completely divisible by 3 if the sum of its digits is divisible by 3 i.e., it is a multiple of 3

Consider a number, 308.To check whether 308 is divisible by 3 or not, take sum of the digits (i.e. 3+0+8= 11). Now check whether the sum is divisible by 3 or not. If the sum is a multiple of 3 then the original number is also divisible by 3. Here, since 11 is not divisible by 3, 308 is also not divisible by 3.

Similarly, 516 is divisible by 3 completely as the sum of its digits i.e. 5+1+6=12, is a multiple of 3.

**Divisibility by 4: **If the last two digits of a number are divisible by 4, then that number is a multiple of 4 and is divisible by 4 completely.

**Example:** Take a number 2308. Consider the last two digits i.e. 08. As 08 is divisible by 4, the original number 2308 is also divisible by 4.

**Divisibility by 5: **Numbers with last digit 0 or 5 are always divisible by 5.

**Example:** 10, 10000, 10000005, 595, 396524850 etc.

**Divisibility by 6: **Numbers which are divisible by both 2 and 3 are divisible by 6. That is, if 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.

**Example:** 630, the number is divisible by 2 as the last digit is 0.

The sum of digits is 6+3+0 = 9, which is also divisible by 3.

Hence 630 is divisible by 6.

**Divisibility by 7: **The rule for divisibility by 7 is given below:

**Example: **Is 1073 divisible by 7?

- From the rule stated remove 3 from the number and double it, which becomes 6.
- Remaining number becomes 107, so 107-6 = 101.
- Repeating the process one more times, we have \(1 \times 2 = 2\)
- Remaining number 10 – 2 = 8.
- As 8 is not divisible by 7, hence the number 1073 is not divisible by 7.

**Divisibility by 8: **If the last three digits of a number is divisible by 8, then the number is completely divisible by 8.

**Example:** Take a number 24344. Consider the last two digits i.e. 344. As 344 is divisible by 8, the original number 24344 is also divisible by 8.

**Divisibility by 9:** 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. **Example: **Consider 78532, as the sum of its digits (7+8+5+3+2) is 25, which is not divisible by 9, hence 78532 is not divisible by 9

**Divisibility by 10: **Divisibility rule for 10 states that any number whose last digit is 0, is divisible by 10.

**Example:** 10, 20,30,1000,5000,60000 etc.

**Divisibility by 11:** If the difference of the sum of alternative digits of a number is divisible by 11 then that number is divisible by 11 completely.

In order to check whether a number like 2143 is divisible by 11 following is the procedure.

- Group the alternative digits i.e. digits which are in odd places together and digits in even places together. Here 24 and 13 are two groups.
- Take the sum of the digits of each group i.e. 2+4=6 and 1+3= 4
- Now find the difference of the sums; 6-4=2
- If the difference is divisible by 11, then the original number is also divisible by 11. Here 2 is the difference which is not divisible by 11.
- Therefore, 2143 is not divisible by 11.

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