I was walking with my girlfriend other day, and I started telling her how Indian mathematics education is quite different and taught quite differently. One example I gave her was that they teaches you that a natural number is divisible by 3 if and only if the sum of its digits (in decimal representation) are also divisible by 3. Then I asked, “do you know why?” without knowing the answer. So both of us started thinking of ways to prove it. After a moment of thoughts, I came up with the proof idea for one direction: if the sum of digits of a natural number is divisible by 3, then the number is divisible by 3. Let’s formally prove this.
Proof that if the sum of digits of a natural number in decimal representation is divisible by 3 then the number is also divisible by 3
Let n be a natural number with the decimal representation where k is a non-negative integer and . Suppose the sum of digits is divisible by 3, which is to say
Then . Because , it follows that by properties of modular arithmetic. Thus . Hence . By the supposition, , and therefore as desired. Q.E.D.
The proof for the other direction, however, quite troubled me to come up with. By supposing that the number is divisible by 3 doesn’t really give you any information about each digit. But this direction can be proved easily using the technique of proof by contradiction.
Proof that if a natural number is divisible by 3, then the sum of its digits in decimal representation is also divisible by 3
Suppose not. Suppose there exists a natural number n that is divisible by 3 and the sum of its digits in decimal representation is not divisible by 3. That is to say if is the decimal representation of n, then . It follows immediately that . But using the same argument made for the previous proof (this result is derived without using the divisibility of the sum of digits in the previous proof). By supposition, so . Thus n is both divisible and not divisible by 3, which is a contradiction. Thus our supposition must be false and there is no natural number divisible by 3 and the sum of its digits in decimal representation is not divisible by 3, proving the statement. Q.E.D.