Find the largest five-digit number, all digits of which are different and which is divisible by 71?

Let’s write down the largest of the five-digit numbers, all of which are different: 98765.

Divide this number with a remainder by 71:

98765/71 = 1391 rest (4).

Since 98765 is a multiple of 71 with a remainder of 4, the number
98765 – 4 = 98761 is divisible by 71 without remainder.

All digits of number 98761 are different and, since there are no other numbers divisible by 71 between this number and the largest five-digit number consisting of different digits, this is the desired number.

Answer: 98761.