How to Prove the divisibility rule for $3$

The divisibility rule for $3$ is well-known: if you add up the digits of $n$ and the sum is divisible by $3$, then $n$ is divisible by three. This is quite helpful for determining if really large numbers are multiples of three, because we can recursively apply this rule:

$$1212582439 \rightarrow 37 \rightarrow 10\rightarrow 1 \implies 3\not\mid 1212582439$$
$$124524 \rightarrow 18 \rightarrow 9 \implies 3\mid 124524$$

This works for as many numbers as I’ve tried. However, I’m not sure how this may be proven. Thus, my question is:

Given a positive integer $n$ and that $3\mid\text{(the sum of the digits of $n$)}$, how may we prove that $3\mid n$?

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HINT: Suppose that you have a four-digit number $n$ that is written $abcd$. Then

$$\begin{align*}
n&=10^3a+10^2b+10c+d\\
&=(999+1)a+(99+1)b+(9+1)c+d\\
&=(999a+99b+9c)+(a+b+c+d)\\
&=3(333a+33b+3c)+(a+b+c+d)\;,
\end{align*}$$

so when you divide $n$ by $3$, you’ll get

$$333a+33b+3c+\frac{a+b+c+d}3\;.$$

The remainder is clearly going to come from the division $\frac{a+b+c+d}3$, since $333a+33b+3c$ is an integer.

Now generalize: make a similar argument for any number of digits, not just four. (If you know about congruences and modular arithmetic, you can do it very compactly.)

$\begin{eqnarray} \rm{\bf Hint}\ \ &&\rm3\ \ divides\ \ a\! +\! 10\,b\! +\! 100\, c\! +\! 1000\,d\! + \cdots\\
\iff &&\rm 3\ \ divides\ \ a\! +\! b\! +\! c\! +\! d\! +\! \cdots +\color{#c00}9\,b\! +\! \color{#c00}{99}\,c\! +\! \color{#c00}{999}\,d\! + \cdots\\
\iff &&\rm3\ \ divides\ \ a\! +\! b\! +\! c\! +\! d + \cdots\ \ by\ \ 3\ \ divides\ \ \color{#c00}{9,\ 99,\ 999,\,\ldots}\end{eqnarray}$

Above we used that $\rm\ n + 3m\ $ is divisible by $\rm\,3\iff n\:$ is divisible by $\,3.$

$1$. First prove that $3 \mid 10^n – 1$ (By noting that, $10^n – 1 = (10-1)(10^{n-1} + 10^{n-2} + \cdots + 1)$).

$2$. Now any number can be written in decimal expansion as
$$a = a_n 10^n + a_{n-1} 10^{n-1} + \cdots + a_1 10^1 + a_0$$

$3$. Note that $a_k 10^k = a_k + a_k (10^k-1)$. Hence,
$$a = \overbrace{(a_n + a_{n-1} + \cdots + a_0)}^{b} + \underbrace{\left(a_n (10^n-1) + a_{n-1} (10^{n-1}-1) + \cdots + a_1 (10^1-1) \right)}_{c}$$

$4$. We have $a=b+c$ and $3 \mid c$. Now conclude that $3 \mid a \iff 3 \mid b$.

How about induction?

It is obviously true for the one-digit numbers $3, 6$ and $9$, so we have our base case (really, just the case $3$ is all it takes, but I like to be on the safe side when it comes to induction).

Now, let’s say that we have a number divisible by $3$, and let’s call it $n$. We can also assume that the sum of the digits of $n$ is divisible by $3$. I want to show that the sum of digits of $n+3$ is also divisible by $3$. If that is the case, then we are done, for the induction principle takes care of any case for us from there.

The sum of the digits of $n$ is some number, let’s call it $m$, and this number is assumed to be divisible by $3$. Now, if we’re lucky, the sum of digits in $n+3$ is just $m+3$, and by lucky I mean there is no carry involved. So, if there is no carry involved in adding $3$ to $n$, then we are done.

If there is a carry, however, then let’s pretend for a second that the last digit of $n$ can surpass $9$. Were that the case, the sum of digits of $n+3$ would really be $m+3$. This is sadly not the case, but what really happens when we do the carry? We subtract $10$ from the $1$-digit, and add $1$ to the $10$-digit. This will have the net effect on the sum of digits that we subtract $9$, so in that case the sum of digits in $n+3$ is $m+3-9 = m-6$, which is still divisible by $3$, so there is no problem!

“Hold on there, not so fast”, you say. “What if adding $1$ to the $10$-s digit makes a carry happen there?” Well, my enlightened reader, in that case the same argument as in the paragraph above would apply, only moved one space to the left in the digits of $n$. The net effect: the sum of digits of $n+3$ is $m-6-9 = m-15$, still divisible by $3$. If there is a carry from the hundreds-digit, then we will subtract another $9$ for a total of $m-24$. And so on. You will never make a carry like that take $m+3$ out of divisible-by-three-space. And this concludes the proof.

More generally, a number and the sum of its digits both leave the same remainder on division by $3$. For example: $245$ $\mapsto 2+4+5=11$ $\mapsto1+1=2$, so the remainder when $245$ is divided by $3$ is $2$.

If you know modular arithmetic, this is straightforward:
\begin{align}
245 & = 2\cdot10^2 +4\cdot10+5 \\[8pt]
& \equiv 2\cdot1^2 + 4\cdot1 + 5 \pmod 3 \\[8pt]
& = 2+4+5 \\[8pt]
& = \text{sum of digits.}
\end{align}

The point is that $10$ is congruent to $1$ when the modulus is $3$, since the remainder when dividing $10$ by $3$ is $1$, and so powers of $10$ are congruent to powers of $1$, and powers of $1$ are just $1$.

Yes, it is true. Let
$$n = a_n a_{n-1} … a_1 a_0 $$
the integer, where $0 \le a_i \le 9$ are its digits, that is
$$ n = \sum_{i=0}^n a_i \cdot 10^i \ . $$
Since $10^i \equiv 1 \pmod 3$ ($10=3 \cdot 3 + 1$, $100 = 3 \cdot 33 + 1$ $1000 = 3 \cdot 333 + 1$ and so on), we can write
$$ n = \sum_{i=0}^n a_i \cdot 10^i \equiv \sum_{i=0}^n a_i \pmod{3} $$
so $n$ is divisbile by 3, if and only if the sums of its digits is
$$ (n \equiv 0 \pmod{3} \iff \sum_{i=0}^n a_i \equiv 0 \pmod{3} )$$
Q.E.D.

There’s likely a more elegant proof, but you got me thinking about this, so here’s what I came up with (this is for two digits, but it generalizes easily).

If some number 10a+b (for a,b integers) is divisible by three then there is some integer k such that
10a+b = 3k

This means that

10a+10b = 3k+9b and thus

10(a+b) = 3(k+3b)

so since 3 doesn’t divide 10, it must divide a+b.

For the other direction:

If a+b = 3k, then

10a+b = 3k+9a = 3(k+3a)

Consider the following example:

Let us have a 3 digit number that can be divided by 3, ie xyz.

Therefore xyz=0 (mod3)

iff xyz=(100x)+(10y)+z=x+y+z=0(mod3)

Therefore x+y+z=0(mod 3), meaning that the sum of the digits is divisible by 3.

This is an if and only if statement.

You can generalize it to n digit numbers. The idea is to express the n digit numbers in powers of 10. Since powers of 10=1 (mod 3), the digit is divisible by 3 iff the sum is divisible by 3.

Divisibility by $3$ rule: $ 3\mid \overline {a_1a_2…a_n} \iff 3\mid (a_1+a_2+…+a_n)$, whereas $a_1,a_2,..a_n$ are digits in $\{0,1,2,…9\}$.

Proof: $\overline{a_1a_2…a_n} = a_1\cdot 10^{n-1} + a_2\cdot 10^{n-2} + … + a_{n-1}\cdot 10 + a_n = a_1\cdot (9+1)^{n-1} + a_2\cdot (9+1)^{n-2} +…+ a_{n-1}\cdot (9+1) + a_n \cong a_1+a_2+…+a_n\pmod 3$.

Example: $3 \mid 4,722$ because $4+7+2+2 = 15$, and $1+5 = 6$ finally is a multiple of $3$. Check: $4,722 = 1,574 \times 3$.

Let $a_k…a_0$ be the digits of $n$. Then

$$n=10^ka_k+\cdots+10a_1+a_0$$
and hence

$$n -\mbox{sum of its digits}=\left( 10^ka_k+\cdots+10a_1+a_0\right)-\left( a_k+\cdots+a_1+a_0\right)\\
=a_k(10^k-1)+\cdots+a_1(10-1)=a_k\cdot 99\ldots9+\cdots+a_1 \cdot 9 \\
=9\left( a_k\cdot 11\ldots1+\cdots+a_2\cdot 11+a_1 \right)$$

As $n -\mbox{sum of its digits}$ is a multiple of nine, it is a multiple of $3$.

Hint: take the number apart into digits. Each digit $d$ represents $d \cdot 10^n$ for some $n$. What is the remainder when you divide $10^n$ by $3$? (Think about $10^n-1$ what does it look like?)

Let abc be a 3 digit number divisible by 3.
Then:
$$(100a+10b+c)|3=0$$
or
$$(100|3)(a|3)+(10|3)(b|3)+(c|3)=0$$
Hence
$$(a+b+c)|3=0$$

Using Property$\#10$ of this ( indirect Proof),

as $10\equiv1\pmod9,10^r\equiv1^r\equiv1$ for integer $r\ge0$

$$\implies\sum_{r=0}^n10^ra_r\equiv\sum_{r=0}^na_r\pmod9$$

Hint: Represent your number as $10^na_n+\cdots 10a_1+a_0$ where the $a_i’s$ are the digit of your number. Now, note that the remained we get when we divide $10^k$ by $3$ is $1$.

Hint $\rm\ mod\ 3\!:\ 10\equiv 1\:\Rightarrow\: n = d_k 10^k+\cdots+d_1 10 + d_0 \equiv d_n+\cdots+d_1+d_0 = $ digit sum.

Or, $ $ let $\rm\:f = \,$ above polynomial (in $10),\:$ so $\rm\ n = f(10),\:$ and $\rm\ f(1) =\,$ digit sum of $\rm\,n.$

Factor Theorem $\rm\:\Rightarrow\: 3\mid 10\!-\!1\mid f(10)\!-f(1)\,$ $\Rightarrow$ $\rm\, f(10) = f(1) + 3k,\ $ so $\rm\ 3\mid f(10)\!\iff\!3\mid f(1)$

Remark $\ $ The same holds true if we replace $\,3\,$ by $\,9\,$ since, too, $\rm\:10\equiv 1\ \ (mod\ 9),\:$ i.e. $\rm\:9\mid 10\!-\!1.\ $ This leads to the casting out nines divisibility test, on which much has been written here.

For $n \in \mathbb{N}$, let $m = \overline{a_0a_1a_2\cdots a_{n-1}}$ be an $n$-digit natural number, where the $a_i$ are digits (natural numbers between $0$ and $9$, inclusive).

Then

$$ m = \displaystyle\sum\limits_{k=0}^{n-1}10^k \cdot a_k = a_0 + 10a_1 + 100a_2 + \cdots + 10^{n-1}a_{n-1}$$

Now consider the equation modulo $3$:

$$ m \equiv \displaystyle\sum\limits_{k=0}^{n-1} 10^k \cdot a_k \pmod{3}$$

$$ m \equiv \displaystyle\sum\limits_{k=0}^{n-1} (9+1)^k \cdot a_k \pmod{3}$$

Since $9 \equiv 0 \pmod{3}$, $9+1 \equiv 1 \pmod{3}$, and since $1^k \equiv 1 \pmod{3}$, we have

$$ m \equiv \displaystyle\sum\limits_{k=0}^{n-1} a_k \pmod{3}$$

This says that the remainder when $m$ is divided by $3$ is the same as the remainder of the sum of the digits of $m$ when that is divided by $3$.

This can be easily proven by “Digital Root” concept.

Digital root: A digit obtained by adding digits of number until a single digit is obtained.

All natural number is partitioned into 9 equivalence class by “Digital root”.

Any number of Digital root 1 is represented by $1+9\times n$
any number of Digital root 2 is represented by $2+9\times n$ and so on.

(the reason for writing like this is: 9 is identity in the case of finding digital root)

So a number whose sum is 3. i.e., Digital root is 3 can be written as $3+9*i$.
Which is divisible by 3.(It’s clear from this representation).