# Smallest Multiple of 9 with all Digits Even

## Theorem

$288$ is the smallest integer multiple of $9$ all of whose digits are even.

## Proof 1

By the brute force technique:

 $\displaystyle 1 \times 9$ $=$ $\displaystyle 9$ $\displaystyle 2 \times 9$ $=$ $\displaystyle 18$ $\displaystyle 3 \times 9$ $=$ $\displaystyle 27$ $\displaystyle 4 \times 9$ $=$ $\displaystyle 36$ $\displaystyle 5 \times 9$ $=$ $\displaystyle 45$ $\displaystyle 6 \times 9$ $=$ $\displaystyle 54$ $\displaystyle 7 \times 9$ $=$ $\displaystyle 63$ $\displaystyle 8 \times 9$ $=$ $\displaystyle 72$ $\displaystyle 9 \times 9$ $=$ $\displaystyle 81$ $\displaystyle 10 \times 9$ $=$ $\displaystyle 90$ $\displaystyle 11 \times 9$ $=$ $\displaystyle 99$ $\displaystyle 12 \times 9$ $=$ $\displaystyle 108$

All integer multiples of $9$ $k \times 9$ for $k = 12$ through to $k = 22$ begin with $1$, and so have at least one odd digit.

Then:

 $\displaystyle 22 \times 9$ $=$ $\displaystyle 198$ $\displaystyle 23 \times 9$ $=$ $\displaystyle 207$ $\displaystyle 24 \times 9$ $=$ $\displaystyle 216$ $\displaystyle 25 \times 9$ $=$ $\displaystyle 225$ $\displaystyle 26 \times 9$ $=$ $\displaystyle 234$ $\displaystyle 27 \times 9$ $=$ $\displaystyle 243$ $\displaystyle 28 \times 9$ $=$ $\displaystyle 252$ $\displaystyle 39 \times 9$ $=$ $\displaystyle 261$ $\displaystyle 30 \times 9$ $=$ $\displaystyle 270$ $\displaystyle 31 \times 9$ $=$ $\displaystyle 279$ $\displaystyle 32 \times 9$ $=$ $\displaystyle 288$

Hence the result.

$\blacksquare$

## Proof 2

Let $n$ be the smallest integer multiple of $9$ all of whose digits are even.

From Divisibility by 9, the digits of $n$ must add to an integer multiple of $9$.

But from Sum of Even Integers is Even, the digits of $n$ must add to an even integer multiple of $9$: $18, 36, 54$, etc.

There is only $1$ integer multiple of $9$ with $2$ digits whose digits add to an even integer multiple of $9$, and that is $99$.

Thus we have to look at $3$-digit integers.

The following sets of $3$ even digits add to $18$:

$\left\{ {2, 8, 8}\right\}$
$\left\{ {4, 6, 8}\right\}$
$\left\{ {6, 6, 6}\right\}$

and that seems to be about it.

There are no sets of $3$ even digits which add up to $36$ or higher.

The result follows by inspection.

$\blacksquare$

## Historical Note

David Wells, in his Curious and Interesting Numbers, 2nd ed. of $1997$, attributes this to an A.J. Turner, but it has not so far been possible to determine who this refers to.

This question seems to be a classic that regularly crops up in compendia of puzzles.