IIT JEE , AIEEE Forum - Mathematics , Algebra - Most of know this but how many can actually prove it ?

ultimator

Prove that a whole number is divisible by 9 if and only if sum of digits is divisible by 9.

ultimator

Come on somebody ! I dont claim that the answer is easy, but u can atleast try.

Surely thr must be some1 who can do this easily.....

GoNik

CALL ALLAMRAJU PLEEEEEEZ

GoNik

I THINK HE CAN DO IT......LIKE OTHER goiitians...but i have more faith in him.....sry..

thedumbheadwithnobrain

TIMEslasher00

it can be proved in not a very convincing way:
Statement 1: The sum of digits of every number which is divisible by 9 comes out to be a multiple of 9.
As this statement is true, so it's converse is also true:
Statement 2: (converse of 1) Every number whose sum of digits is a multiple of 9, must be divisible by 9,.

Decoder

nice methods...abv..

here i go with a general proof..( very unorthodox..bas aise hi aa gaya)..

let f(x) = ax^n + bx^n-1 .....+.z...(n be any natural no.)

now we know tht... f(10) mod 9 = 0...

=> f(9+1) mod 9 = 0...

now by binomial expansion ....

we get f(10) = a(9 + 1)^n. + b (9+1)^n-1...+..z ...

=>> 9m + a + b + c + d.....+z...

since l.hs is a multiple of 9...

so( a + b + c + ...+z) is also divisible by 9...

hence sum of digits or f(1) is also divisible by 9...

oneyeartogo

A comparatively more difficult problem, but easy on the whole.

Prove that a whole number is divisible by 11 if difference of sum of alternate digits is a multiple of 11.

You can try proving the rest of the divisibility tests. They r very easy to do.

sandeepramesh

Its a direct consequence of modulos! We all know that a number can be represented by the form $\sum_{i=0}^{n} 10^{i} \cdot a_{i}$ where $a_{i} \in \mathbb{R^{+}}$ say

Ofc $10 \equiv 1 \mod 9$

Hence the result Razz Mr. Green

little_genius

let n= ....abcd = (.....+ 1000a+ 100b+10c + d )/9= Q + (a+b+ c+d)/9....
hence for it to b divisible by 9 the sum of digits shud b divisible by 9..

shortest way:))

hsbhatt

So many posts on this topic is overkill in my opinion

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