Iff a number $n$ is divisible by $3$, then the sum of its digits is also divisible by $3$.

You are watching: A number is divisible by 3 if the sum of its digits is divisible by 3

Proof:

We know $n \mod 3 = 0$. By the basis representation theorem, $n$ can be re-written as $n_k10^k + n_{k-1}10^{k-1} + \cdots + 10n_1 + n_0 \equiv 0 \mod 3$. By the modular arithmetic addition rule, we can take $n_k10^k \mod 3 + n_{k-1}10^{k-1} \mod 3 + \cdots + 10n_1 \mod 3 + n_0 \mod 3$. But then we just get $n_k + n_{k-1} + \cdots + n_1 + n_0$, since $10 \mod 3 \equiv 1$.

Now suppose that the sum of the digits was divisible by 3. How can I prove that the representation in base $10$ is divisible by $3$?

elementary-number-theory

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asked Jan 17 "15 at 23:44

Don LarynxDon Larynx

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## 2 Answers 2

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The key point which you seem to have noted is that $10\equiv 1\pmod{3}$. This means that a number is congruent to the sum of its digits mod 3 because $10^k\equiv 1\pmod{3}$, which you also seem to have noted. Thus $n$ is divisible by $3$ (congruent to $0$ mod 3) if and only if the sum of its digits is. The same is true for other remainders modulo 3; a number has a remainder of $1$ when divided by $3$ if and only if the sum of its digits does, etc.

Your observation that $10^k\equiv 1\pmod{3}$ is the reason why it has been commented that you have already proved the result.

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answered Jan 17 "15 at 23:56

Matt SamuelMatt Samuel

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I hope this will suffice:$$n= (d_{m-1}\cdots d_0)_{10} = \sum_{k=0}^{m-1} d_k 10^k= \sum_{k=0}^{m-1} d_k (10^k - 1) + d_k = \sum_{k=0}^{m-1} d_k (10^k - 1) + S(n)$$with $S(n)$ being the digit sum in base $10$ and$$10^k - 1 = 9 \frac{10^{k} - 1}{10-1} = 9 \sum_{i=0}^{k-1} 10^i = \sum_{i=0}^{k-1} 9 \, 10^i= (\underbrace{9\cdots 9}_{k \times})_{10} $$is divisible by $3$, it depends on the divisibility of $S(n)$.

The same statement should hold for division by $9$.

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edited Jan 18 "15 at 0:12

answered Jan 17 "15 at 23:55

mvwmvw

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