Smallest Multiple of 9 with all Digits Even

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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.


Sources