Question

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

Answer 1

`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+color(blue)(1) = 15` is divisible by `3`.

So `1364color(blue)(1)` is divisible by `3`, and so will be:

`1364color(blue)(1)+3 = 1364color(blue)(4)`

and

`1364color(blue)(4)+3 = 1364color(blue)(7)`

`color(white)()`
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:

`color(red)(153) = (100*color(red)(1)) + (10*color(red)(5)) + (1*color(red)(3))`

`= (99+1)*color(red)(1) + (9+1)*color(red)(5) + (0+1)*color(red)(3)`
`= (99*color(red)(1) + 9*color(red)(5) + 0*color(red)(3)) + (1*color(red)(1) + 1*color(red)(5) + 1*color(red)(3))`
`= 3(33*color(red)(1) + 3*color(red)(5) + 0*color(red)(3)) + (color(red)(1)+color(red)(5)+color(red)(3))`

The expression `3(33*color(red)(1)+3*color(red)(5)+0*color(red)(3))` is divisible by `3`.

So we find that `153` is divisible by `3` if and only if `(1+5+3)` is.