# 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:

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

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:

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

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 $18$:

$\set {2, 8, 8}$
$\set {4, 6, 8}$
$\set {6, 6, 6}$

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.