Information about Beautiful Number 9

Magic of Number 9

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.

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.

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.

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.

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