What is the largest 4-digit number, with all different digits, that is divisible by each of its digits?

 

 

The largest number fitting the description is 9876.

 

To check for divisibility by 9, add the digits.  If the sum of the digits is divisible by 9, the number is.

 

, not divisible by 9.

 

The first smaller number that is divisible by 9 will be 3 less than the given number, or 9873.

 

 so this number is divisible by 9.

 

To check divisibility by 8, we see if the last three digits are divisible by 8.

 

873 is odd, so it cannot be divisible by 8.

 

The next smaller number divisible by 9 is 9 less than 9873 or 9864.

 

864 is divisible by 8.

 

To check for divisibility by 6, we see if it is divisible by both 2 and 3.  Since it is divisible by 9 it is divisible by 3.  Since it is even it is divisible by 2.  So it is divisible by 6.

 

To check divisibility by 4, we see if the last two digits are divisible by 4.  64 is divisible by 4.

 

9864 meets the requirements.