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What is the smallest positive integer, which has - for each of the digit 0-9 - a divisor ending with this digit?

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It's

$270$, which has divisors $1, 2, 3, 54, 5, 6, 27, 18, 9, 10$.

Note that

the number must be a multiple of $10$

and then just

try every multiple of $10$ until it hits.

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  • $\begingroup$ Because 10 is coprime with with any number ending in 9 then you only need to check multiples of 90,190,290,... which significantly reduces the search space. $\endgroup$
    – hexomino
    Apr 11 at 8:29

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