# If the 5 digit number 1364"?" is divisible by 3 then what are the possible values of the last digit?

Jun 11, 2016

$1$, $4$ or $7$.

#### Explanation:

In the decimal number system that we use, an integer is divisible by $3$ if and only if the sum of its digits is also divisible by $3$.

$1 + 3 + 6 + 4 + \textcolor{b l u e}{1} = 15$ is divisible by $3$.

So $1364 \textcolor{b l u e}{1}$ is divisible by $3$, and so will be:

$1364 \textcolor{b l u e}{1} + 3 = 1364 \textcolor{b l u e}{4}$

and

$1364 \textcolor{b l u e}{4} + 3 = 1364 \textcolor{b l u e}{7}$

$\textcolor{w h i t e}{}$
Footnote

Why does this method of checking the digits add up to a multiple of $3$ work?

Essentially because when you divide $10$ by $3$ then the remainder is $1$.

So for example:

$\textcolor{red}{153} = \left(100 \cdot \textcolor{red}{1}\right) + \left(10 \cdot \textcolor{red}{5}\right) + \left(1 \cdot \textcolor{red}{3}\right)$

$= \left(99 + 1\right) \cdot \textcolor{red}{1} + \left(9 + 1\right) \cdot \textcolor{red}{5} + \left(0 + 1\right) \cdot \textcolor{red}{3}$
$= \left(99 \cdot \textcolor{red}{1} + 9 \cdot \textcolor{red}{5} + 0 \cdot \textcolor{red}{3}\right) + \left(1 \cdot \textcolor{red}{1} + 1 \cdot \textcolor{red}{5} + 1 \cdot \textcolor{red}{3}\right)$
$= 3 \left(33 \cdot \textcolor{red}{1} + 3 \cdot \textcolor{red}{5} + 0 \cdot \textcolor{red}{3}\right) + \left(\textcolor{red}{1} + \textcolor{red}{5} + \textcolor{red}{3}\right)$

The expression $3 \left(33 \cdot \textcolor{red}{1} + 3 \cdot \textcolor{red}{5} + 0 \cdot \textcolor{red}{3}\right)$ is divisible by $3$.

So we find that $153$ is divisible by $3$ if and only if $\left(1 + 5 + 3\right)$ is.