Take a 4 digit number, say n=1234.

Only 6 of the 24 numbers created by permuting
the digits of n

are divisible by 4: 1324, 1432, 3124, 3412, 4132, 4312.

Let us denote by r(1234) the ratio 6/24=1/4 .

Clearly r(1034)=1/6, since only 1340,1304,3104 and 3140 qualify.

a. What is the highest ratio r achievable for a 4 digit number?

b. How many 4 digit numbers have this ratio?

c. Questions a and b for 5 digit numbers, re division by 4 as well.

Rem:

In questions a-c the digits must be distinct - one leading zero allowed.