Every Digit, Every Divisor

1. Find the smallest number that uses each of the ten digits $0$ to $9$ once and is divisible by every one of $1, 2, 3, \dots, 9$.

2. Find the smallest such ten-digit number divisible by every whole number from $2$ to $18$.

Solution

To be divisible by all of $1$ to $9$ is to be divisible by their lowest common multiple, which is $2520$. So we hunt for the smallest ten-digit number, using each digit once, that is a multiple of $2520$. It is $$1234759680 = 2520 \times 489984.$$ For the second part the lowest common multiple of $2$ through $18$ is $12252240$, and the smallest pandigital multiple of it is $$2438195760 = 12252240 \times 199.$$ A small bonus: every number using all ten digits once is automatically divisible by $9$, because its digits add to $0 + 1 + \dots + 9 = 45$, itself a multiple of $9$.