Question

Show that an integer is divisible by 9if and only if the sum of its decimal digits is divisible by9

Short answer

m is divisible by 9 if and only if 9 divides the sum of its decimal digits $\left(a_0+\dots ..+a_n\right)$

Step-by-step solution

Step 1

Let m be an integer. Writing its decimal expansion.

$m=a_0\ast {10}^0+a_1\ast {10}^1+\dots ..+a_n\ast {10}^n=a_0\ast (9+1{)}^0+a_1\ast (9+1{)}^1+\dots .+a_n\ast (9+1{)}^n=a_0+a_1+\dots +a_n+9k$

for some integer k .

We have used the following fact -

Given any positive integer , for some integer .(can be proved by induction).

So $m=a_0+a_1+\dots .+a_n+9k$.

Hence m is divisible by 9 if and only if 9 divides the sum of its decimal digits .

$\left(a_0+\dots .+a_n\right)$