Henry Ernest Dudeney/Modern Puzzles/63 - Find the Factors/Solution

Modern Puzzles by Henry Ernest Dudeney: $63$

Find $2$ whole numbers with the smallest possible difference between them

which, when multiplied together, will produce $1234567890$.

$36070 \times 34227$

We have:

$1234567890 = 2 \times 3^2 \times 5 \times 3607 \times 3803$

It is safe to say that one of the factors will contain $3607$ and the other $3803$.

It remains a matter of putting these together with the smaller ones to get the smallest difference.

The remaining factors can be collected as $3^2 = 9$ and $2 \times 5 = 10$

$\ds 1234567890$

$=$

$\ds \paren {9 \times 3803} \times \paren {10 \times 3607}$

and so to the result.

$\blacksquare$