Explain the divisibility rules.

Divisibility rules:

Divisibility rule of 1.

Every number is divisible by 1.

Any number divided by 1 will give the number itself.

For example,

5 is divisible by 1 and 5000000 is also divisible by 1.

Divisibility rule of 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.

For example,

68462 is an even number and divisible by 2 but 48465 is not an even number, hence not divisible by 2.

Divisibility Rules of 3

Divisibility rule for 3 states that a number is completely divisible by 3 if the sum of its digits is divisible by 3.

For example,

405 is divisible by 3. Because the sum of the digits $4+0+5=9$ is divisible by 3.

Divisibility rule of 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.

For example,

Take the number 4016. Consider the last two digits, 16. As 16 is divisible by 4, the original number 4016 is also divisible by 4.

Divisibility rule of 5

Numbers with the last digit 0 or 5 are always divisible by 5.

For example,

50, 100, 100005, 50055, 6546460 are all divisible by 5.

Divisibility rule of 6

Numbers that are divisible by both 2 and 3 are divisible by 6.

That is, 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.

For example,

786

The number is divisible by 2 as the last digit is 6.

The sum of digits is $7+8+6 = 21$, which is also divisible by 3.

Hence 786 is divisible by 6.

Divisibility Rules of 7

Cross off the last digit, double it and subtract. Repeat if you want. If the new number is divisible by 7, the original number is divisible by 7.

For example,

875

$87-(2\times5)=87-10=77$.

77 is divisible by 7.

Therefore, 875 is divisible by 7.

Divisibility rule of 8

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

For example,

48688.

The last three digits are 688.

688 is divisible by 8, the original number 48688 is also divisible by 8.

Divisibility by 9

The rule for divisibility by 9 is similar to the 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.

For example,

64654,

The sum of the digits $(6+4+6+5+4)$ is 25, which is not divisible by 9, hence 64654 is not divisible by 9.

Divisibility rule of 10

The divisibility rule for 10 states that any number whose last digit is 0, is divisible by 10.

For example,

10, 20,30,100,8000,30000 etc.

Divisibility Rules of 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.