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Pandigital Multiple (Posted on 2011-11-25) Difficulty: 3 of 5
Determine the probability (in terms of n) that a base-n positive integer, starting with a nonzero digit and containing each of the digits 0, 1, 2,..., n-1 exactly once, is a multiple of the base-n number 11.

No Solution Yet Submitted by K Sengupta    
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Some Thoughts Sometimes impossible, not nearly solved | Comment 1 of 2
I'm nowhere near having solved this but I can show that in some cases it is not possible.  It appears to me that any formula will be extremely complicated.

The divisibility rule for 11 on base 10 is the same as in any base, n.

If the difference between the sums of the numbers in the even and odd places is a multiple of 11, the number is divisible by 11.

So we can think of the pandigital number is being split into its even and odd places and call these sums a and b (a>b).

In base n, the digits 0 through (n-1) sum to n(n-1)/2 = nē/2 - n/2.
In base n, the number 11 is equal to n+1
so we have the system

a+b = nē/2 - n/2
a-b = k(n+1)

Solving for b gives b = (nē-n)/4 - k(n+1)/2

In different bases this may not be possible:
Base 2: b= .5-1.5k does not give a positive integer for non-negative k

Base 3: b=1.5 - 2k never gives an integer

Base 4: b= 3 - 2.5k if k=0, b=3 (and a=3) which gives 6 possibilities including for example 3201

Base 5: b = 5 - 3k which gives two possibilities k=0, b=5 (example: 43120) and k=1, b=2 (ex: 40321)

Base 6: b = 7.5 - 3.5k which gives k=1, b=4 (ex: 534120)

Base 7: b = 10.5 - 3k never gives an integer

Base 8: b = 14 - 4.5k which gives k=0, b=14 (ex: 76453201) and  k=2, b = 5 which is impossible since the smallest 4 digits sum to 6

Base 9: b=18-5k which gives k=0, b=18 (many), k=1, b=13 (many), k=2, b=8 (ex: 857251403), k=3, b=3 is too small

Base 10, b = 22.5 - 5.5k which gives k=1, b=17 (ex: 8071624539), k=3, b=6 which is too small.

I haven't the time to decide for which larger n it will be impossible.  

Just to count the base 10 solutions you would need to find every partition of 5 numbers and a sum of 17.  Then for each multiply this by 5! and by 5! excepting some that will cause a leading 0.

  Posted by Jer on 2011-11-28 11:46:24
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