Smallest Multiple of 9 with all Digits Even/Proof 1

Theorem

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

Proof

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.

$\blacksquare$