Smallest Multiple of 9 with all Digits Even/Proof 1
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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.
$\blacksquare$