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 Maximizing the ratio (Posted on 2013-04-18)
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.

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 All parts (spoiler?) Comment 1 of 1
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)

Part b) There are 2 such numbers X 24 permutations each = 48.

Part c) I expected that the maximum 5 digit result is only achieved with 0, 2, 4, 6 and 8.  This has 120 permutations.   This gives a 60% divisibility by 4 (whenever the last digit is not 2 or 6)

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

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