Largest number divisible by its digits

Largest number divisible by its digits is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

Task

Find the largest base 10 integer whose digits are all different, that is evenly divisible by each of its individual digits.

For example: 135 is evenly divisible by 1, 3 and 5.

Note that the digit zero (0) can not be in the number as integer division by zero is undefined. The digits must all be unique so a base 10 number will have at most 9 digits.

Feel free to use analytics and clever algorithms to reduce the search space your example needs to visit, but it must do an actual search. (Don't just feed it the answer and verify it is correct.)

Stretch goal

Do the same thing for hexadecimal.