Divisibility by ‘9’: - A natural number is divisible by ‘9’, if and only if sum of the digit is divisible by ‘9’.
eg:- 1) 306019 = 2 + 0 + 6 + 0 + 1 + 9 = 18 is divisible by 9, then 206019 is also divisible by ‘9’ (Division try yourself)
Note :- 872645 is not divisible by ‘ 9’ as 8 + 7 + 2 + 6 + 4 + 5 = 32 is not divisible by ‘9’
8, 72, 645 is also not divisible by ‘9’.
Divisibility by 10:- A natural number is divisible by 10, if and only if the last digit is zero.
eg:- 2070 is divisible by 10 as 2070 = 207 10
10 is divisible by 10
2070 is also divisible by 10
Divisibility by 11 :- A natural number is divisibile by ‘11’, if and only if, the difference (D) of the sum of the numbers obtained on adding the alternate digits of the numbers seperately (S1 and S2) is divisible by ‘11’ (or) zero.
1) If D = S1-S2 = 0, then the given number is divisible by 11
2) If D = S1 - S2 = multiple of ‘11’ then also ‘D’ is divisible by ‘11
eg:- 1) 4334
4+3= 7 = S1, 3+4=7 = S2
S1-S2 = 7-7 = 0.
4334 is divisible by ‘11’
2) 66311 is not divisible by 11 as
S1 = 6+3+1 = 10, S2 = 6+1 = 7
D = S1-S2 =10 - 7 = 3
clearly ‘3’ is not multiple of 3