Rule of Divisibility – Number System

In a number of situation, we will to find factors of a given number. Some of the factors of a given number can be found very easily either by observation or by applying simple rules.

We will look at some rules for divisibility of numbers:

Divisibility by 2:

A number divisible by 2 will have an even number as its last digit.

In other words, if unit digit of number is 0, 2, 4, 6, 8 ,then number is divisible by 2.

For example, 128, 246, 2346, etc.

Divisibility by 3:

A number is divisible by 3, if the sum of its digits is a multiple of 3.

For example, take the number 8469, the sum of the digit is 8+4+6+9=27 which is a multiple of 3. Hence, the given number 8469 is divisible by 3. Similarly 342, 789, 333, etc

We take another number 74549, the sum of the digit is 29 which is not multiple of 3. Hence the number 74549 is not divisible by 3.

Divisibility by 4:

A number is divisible by 4, if the number formed with its last two digits is divisible by 4.

For example, take the number 954672 the last two digits is 72 which is divisible by 4. Hence the number 954672 is divisible by 4.

We take another number 683278, the last two digits are 78 which is not divisible by 4. Hence the number 683278 is not divisible by 4.

Divisibility by 5:

A number is divisible by 5, if its last is 5 or zero.

For example, 15, 40, 890 etc. all are divisible by 5.

Divisibility by 6:

A number is divisible by 6, if last digit of the number is divisible by 2 and sum of total all digit of the number is divisible by 3.

In other words, A number is divisible by 6 then it is divisible by both 2 and 3.

For example, 18, 96, 9612 etc. all are divisible by 6.

Divisibility by 7:

A number is divisible by 7, if the difference between the number of tens in the number and twice the units digit is divisible by 7, Otherwise, it is not divisible by 7.

For example, take the number 795. The unit digit is 5 and its doubled is 10. The remaining part of the number is 79. If 10 is subtracted from 79 we get 69. Since this result is not divisible by 7. so original number 795 is not divisible by 7.

Divisibility by 8:

A number is divisible by 8, if the number formed by the last 3 digits of the number is divisible by 8.

For example, 128, 34568, 76232, etc. all are divisible by 8.

Divisibility by 9:

A number is divisible by 9 if the sum of its digits is a multiple of 9.

For example, take the number 129835782 and the sum of all its digit is 1+2+9+8+3+5+7+8+2=45 which is divisible by 9. So, number is divisible by 9.

Divisibility by 10:

A number is divisible by 10, if number should be end with zero.

For example, 580, 839480, 32489582450, etc.

Divisibility by 11:

A number is divisible by 11, if the sum of the alternate digits is the same or they differ by multiple of 11.

In other words, the difference between the sum of digits in an odd place in the number and the sum of the digits in the even places in the number should be equal to zero or a multiple of 11.

For example, 6595149 is divisible by 11 as the difference of 6+9+1+9=25 and 5+5+4=14 is 11.

Divisibility by 12:

A number is divisible by 12, if the number is divisible by 3 and 4 both.

For example, 144, 348, 5496, 65952, etc.

Divisibility by 13:

If number is in form (A+4B), where B is the unit place digit and A is remaining digits of number.

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