Find a six digit integer Find an integer with six different digits such that the six digit integer is divisible by each of its digits.
For example, find ABCDEF such that A, B, C, D, E and F all can divide the number ABCDEF.
Show your answer along with mathematical reasoning.

$4\\
24\\
624\\
3624\\
183624$
2,4,8 are good to work with, because once you find three digits at the end, you can put whatever you want in the front.
3,6,9 are good to work with too, because if you find something that divides by 3 or 9, you can shuffle the digits and it still divides by 3 (or 9).
and 1 is just a no-brainer
