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# DIVISIBILITY RULES FOR A NUMBER

Before I start writing divisibility rules we must be familiar with what is the meaning of divisibility. A number is said to be divisible by the numbers that divide it without leaving a remainder. That means if the remainder is 0 then the number is said to be divisible.

## DIVISIBILITY BY 2

A number is said to be divisible by 2 when it is an Even number. A number is said to be Even number when ones digit of the number is any one from 0, 2, 4, 6 or 8. We can say that 36, 68, 112, 1150 and 56894 are all divisible by 2. Because each of them are Even.

## DIVISIBILITY BY 3

A number is said to be divisible by 3 if the sum of its digits is divisible by 3. The sum of digits of 12345 is 1 + 2 + 3 + 4 + 5 = 15. As 15 is divisible by 3. Therefore 12345 is divisible by 3. The sum of digits of 433 is 4 + 3 + 3 = 10. As 10 is not divisible by 3. Therefore 433 is not divisible by 3.

## DIVISIBILITY BY 4

A number is said to be divisible 4 if the number formed by its last two digits is divisible by 4. The number form by the last two digits of 22348 is 48. 48 is divisible by 4. Therefore 22348 is divisible by 4. 1242 is not divisible by 4 because 42 is not divisible by 4.

## DIVISIBILITY BY 5

A number is said to be divisible by 5 if it end with either 5 or 0. For example 55, 170, 12345 And 432560. We can see each number ends with either 5 or 0 therefore each of them is divisible by 5.

## DIVISIBILITY BY 6

A number is said to be divisible by 6 if it is divisible by 2 and 3 both. Let’s check 4560 is divisible by 6 or not. Since 4560 is an even number therefore it is divisible by 2. Now the sum of digits of 4560 is 4 + 5 + 6 + 0 = 15. As 15 is divisible by 3 therefore 4560 is divisible by 3 too. Thus 4560 is divisible 6 because it is divisible by 2 as well as 3. Now check divisibility of 214 by 6. Since 214 is an even number therefore it is divisible by 2 but sum of digits of 214 is 7. As 7 is not divisible by 3 therefore 214 is not divisible by 3. Hence 214 is not divisible by 6. Similarly 321 is divisible by 3 but not divisible 2 therefore 321 is not divisible 6.

## DIVISIBILITY BY 7

A number is said to be divisible by 7 if the difference between the double the last digit and the rest of number is divisible by 7. Let’s check divisibility of 1792 by 2. Double the last digit is 4. The rest of number is 179. Therefore difference is 179 – 4 = 175. Now again checking is 175 divisible by 7? Double the last digit is 10 and rest of number is 17. Therefore difference is 17 – 10 = 7. As 7 is divisible by 7. Thus 175 is divisible by 7. Hence 1792 is divisible by 7.

## DIVISIBILITY BY 8

A number is said to be divisible by 8 if the number formed by its last three digits is divisible by 8. For example the number formed by the last three digits of 15824 is 824. By dividing 824 by 8 we find that it is divisible. Thus 15824 is also divisible by 8. 6324 is not divisible by 8 because the number 324, formed by last three digits of 6324, is not divisible by 8.

## DIVISIBILITY BY 9

A number is said to be divisible by 9 if the sum of digits of the number is also divisible by 9. For example sum of digits of 9972 is 9 + 9 + 7 + 2 = 27. As 27 is divisible by 9 therefore 9972 is also divisible by 9. Another example is as: the number 69226 is not divisible by 9 because the sum of digits of 69226 is 25 and it is not divisible by 9.

## DIVISIBILITY BY 10

A number is said to be divisible by 10 if its ones digit is 0. For example 50, 340, 1720, 55500 and 1520000 are all divisible by 10.

## DIVISIBILITY BY 11

A number is said to be divisible by 11 if the difference of sum of digits of odd places and even places is divisible by 11. Let’s check Is 13244 divisible by 11 or not ?

The sum of digits in odd places is 1 + 2 + 4 = 7 and the sum of digits in even places is 3 + 4 = 7. Now their difference is 7 – 7 = 0. Since 0 is divisible by 11 therefore 13244 is also divisible by 11.

## DIVISIBILITY BY 12

A number is said to be divisible by 12 if it is divisible by 3 as well as 4. For example: Is 1560 is divisible by 12? At first we need to check that is 1560 divisible by 3. Since the sum of digits of 1560 is 12 which is divisible by 3. Thus 1560 is divisible by 3. Now we need to check that is 1560 is divisible by 4. The number formed by the last two digits of 1560 is 60 and it is divisible by 4. Hence 1560 is divisible by 12.

## DIVISIBILITY BY 15

A number is said to be divisible by 15 if it is divisible by 3 and 5 both. For example: Is 1275 divisible by 15? Since sum of digits of 1275 is 1 + 2 + 7 + 5 = 15. As 15 is divisible by 15 therefore 1275 will also be divisible by 3. The number 1275 ends in 5 therefore it is divisible by 5 too. Hence 1275 is divisible by 15.

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