3. Suppose that a positive integer is written in decimal notation as n = akak-1… a2a1a0 where 0 ai 9. Prove that n is divisible by 9 if and only if the sum of its digits ak + ak–1 + … + a1 + a0 is divisible by 9.

Therefore n is divisible by 9 if and only if   is also divisible by 9.

Step-by-step explanation:

Given number is

This means

Here we need to prove

is divisible by 9.

We know that

10  ≡ 1 mod 9

It means if 10 divides by 9 the remainder = 1.

mod 9

mod 9

Therefore n is divisible by 9 if is also divisible by 9.

And conversely is also true.

Therefore n is divisible by 9 if and only if   is also divisible by 9.

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step-by-step explanation: