# 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 $18$:

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

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.

## Sources

- 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $288$