Magic of Number 9

1. Beautiful Number 9 – Part I
•
It is the 3rd square number and 32 = 9.
•
N x 10i – N is always divisible by 9 for any N and any i.
•
A number is divisible by 9 if and only if sum of its digits is
divisible by 9.
•
Take any number, reverse the digits of the number to form a
new number and then subtract the smaller from the larger.
The result is always divisible by 9.
•
Take any number and add 9 with it, the digital root of the
number is always equal to the number itself.

2. A number is divisible by 9 if and only if sum of its digits is
divisible by 9.
Let nk…n3n2n1 be the integer divisible by 9.
To prove that, S = nk + … + n3 + n2 + n1 is also divisible
by 9.
Note that,
nk…n3n2n1 = nk.10(k-1) + … + n3.102 + n2.10 + n1
S = nk
+ … + n3
+ n2
+ n1
N x 10i – N is always divisible by 9 for any N and any i,
so, it clearly follows that nk…n3n2n1 - S is divisible by 9.
Therefore, if nk…n3n2n1 is divisible by 9, then so is S and
vice versa.

3. Take any number, reverse the digits of the number to form
a new number and then subtract the smaller from the
larger. The result is always divisible by 9.
Example – (i) 156 on reversing the digits yield 651
=> 651 – 156 = 495 is divisible by 9.
(ii) 4291 on reversing the digits yield 1924
=> 4291 – 1924 = 2367 is divisible by 9.
Does not it follow from the same reasoning –
N x 10i – N is always divisible by 9 for any N and any i.

4. Digital Root of a Number N is defined as
d(N) =
N mod(9) , if N is not divisible by 9
9
, if N is divisible by 9
It is also obtained by taking the sum of the digits of
the number and then adding the digits of the derived
number and continue in this way until single digit
number is formed. Single digit number thus obtained
is the Digital Root of N.
Example –
(i) d(154) = 1, since 1 + 5 + 4 = 10 and then 1 + 0 = 1.
(ii) d(4199) = 5, since 4 + 1 + 9 + 9 = 23 and then 2 + 3 = 5.

5. Take any number and add 9 with it, the digital root of the
number is always equal to the number itself.
If N is divisible by 9, then N + 9 is also divisible by 9.
If N is not divisible by 9, then N mod (9) = (N + 9) mod
(9)
Hence, d(N) = d(N + 9)
Example –
(i) d(154) = 1 and also d(154 + 9) = d(163) = 1.
(ii) d(4199) = 5 and also d(4199 + 9) = d(4208) = 5.