**Divisibility by 2:**

*A number is divisible by 2, if its unit is any of 0, 2, 4, 6, 8.*

**Divisibility by 3:**

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

**Ex.**Which of the following number is divisible by 3?

(i) 541326 (ii) 5967013

**Sol:**Sum of digits in 541326 is 21, which is divisible by 3. So, 541326 is divisible by 3.

Sum of digits in 5967013 is 31, which is not divisible by 3. So 5967013 is divisible by 3

Hence 5967013 is not divisible by 3.

**Divisibility by 9:**

*A number is divisible by 9. If the sum of its digit is divisible by 9.*

**Ex.**Which of the following numbers is divisible by 9 ?

(i) 19725462 (ii) 36870521

**Sol:**Sum of digits in 19725462 is 36, which is divisible by 9. So, 19725462 is divisible by 9

Sum of digits in 36870521 is 32, which is not divisible by 9.

So, 36870521 is not divisible by 9.

**Divisibility by 4:**A number is divisible by 4, if the number formed by last two digits is divisible by 4.

**Ex.**Which of the following numbers is divisible by 4

(i) 67920594 (ii) 618703572

**Sol:**The number formed by the last two digits in 67920594 is 94, which is not divisible by 4.

:. 67920594 is not divisible by 4.

The number formed by the last 2 digits in 618703572 is 72, which is divisible by 4.

:. 618703572 is divisible by 4.

**Divisibility by 8 :**

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

**Ex.**Which of the following numbers is divisible by 8 ?

(i) 98016542 (ii) 106598304

**Sol.**The number formed by the last 3 digits of 98016542 is 542. Which is not divisible by 8.

:. 98016542 is not divisible by 8.

The number formed by the last 3 digits of the given number 304, which is divisible by 8.

:. 106598304 is divisible by 8.

**Divisibility by 11:**

*A number is divisible by 11. If the difference of the sum of its digits at odd places and the sum of digits at even places, is either 0 or a number divisible by 11.*

**Ex.**Show that 4832718 is divisible by 11.

**Sol.**(Sum of digits at odd places) + (Sum of digits at even places)

[(8 + 7 + 3 + 4) - ( 1 + 2 + 8)] = 11. which is divisible by 11

:. 4832718 is divisible by 11

**Co-Prime:**Two numbers are said to be co-prime, if their H.C.F is 1.

e.g. (2, 3), (4, 5), (7, 9), (8, 11) are co-primes.

Note: A number is divisible by ab only when that number is divisible by each one of a and b, where a and b are co-primes

**Ex.**Without actual division show that 52563744 is divisible by 24.

**Sol.**24= 3x8, where 3 and 8 are co prime.

The sum of the digits in given number is 36, which is divisible by 3.

So, the number is divisible by 3.

The number formed by the last 3-digits of the given number is 744, which is divisible by 8.

So the given number is divisible by 8.

Thus, the given number is divisible by both 3 and 8, where 3 and 8 are co-prime

So, it is divisible by 3x8, i.e. 24