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.

Part a) On quick reflection, it seems that the maximum is likely achieved with 0, 4, and 8 (all divisible by 4) and then another even number (2 or 6).  This gives a 75% divisibility 4 (whenever the last digit is not 2 or 6)

Posted by Steve Herman on 2013-04-18 13:03:28