A number divisible by 2 will have an even number as its last digit (For example 158, 45246 etc)

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

For example, take the number 12318, the sum of the digits is 1+ 2 + 3 + 1 + 8 = 15 which is a multiple of 3. Hence, the given number 12318 is divisible by 3.

A number is divisible by 4 if the number formed with its last two digits is divisible by 4. For example, if we take the number 230564, the last two digits form 64. Since this number 64 is divisible by 4, the number 230564 is divisible by 4.

If we take the number 22222… n   times, the last two digits form 22 which is not divisible by 4 and hence the number 22222… n   times  can never be a multiple of  4.

A number is divisible by 5 if its last digit is 5 or zero (eg. 2015, 1000, etc.)

A number is divisible by 6 if it is divisible both by 2 and 3 which means last place is even and sum of the digits is multiple of 3.

First form the groups of 3 digits each staring from the right of the number. If the difference of sum of the even groups and the sum of the odd groups is zero or multiple of 7, then the number is also multiple of 7. For example if the number is  24013437, the required groups are  \(\underline {024} \,\,\underline {013} \,\,\underline {437} \)

first group is 437 

second group is 013

third group is 024 

Sum of the even groups = 013

Sum of the odd groups = 437 + 024 = 461

Difference =461 – 013 = 448, that is multiple of 7, hence the number 24013437 is multiple of 7.

A number is divisible by 8 if the number formed with its last three digits is divisible by 8. For example, if we take the number 2130512, the last three digits form 512. Since this number 512 is divisible by 8, the number 2130512 is divisible by 8.

Number is divisible by 9 if the sum of its digits is multiple of 9.

For example, if we take the number 12345678, the sum of the digits of this number is 36. Since this sum is a multiple of 9, the number 12345678 is divisible by 9.

A number is divisible by 11 if the sum of the alternate digits is the same or they differ by multiples of 11. For example, if we take the number 1331, the sum of the digits in odd places is 1 + 3 = 4 and the sum of the digits in even places is 4. Since these two sums are equal, the given number is divisible by 11

First form the groups of 3 digits each staring from the right of the number. If the difference of sum of the even groups and the sum of the odd groups is zero or multiple of 13, then the number is also multiple of 13. For example if the number is  24006437, the required groups are  \(\underline {024} \,\,\underline {006} \,\,\underline {437} \)

first group is 437 

second group is 006

third group is 024 

Sum of the even groups = 006

Sum of the odd groups = 437 + 024 = 461

Difference = 461 – 006 = 455, that is multiple of 13, hence the number 24006437 is multiple of 13.