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