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Pandigital Divisibility (Posted on 2010-09-12) Difficulty: 3 of 5
N is a seven digit base-14 positive integer using the digits 1 to 7 exactly once.

Determine the total number of value(s) of N that are divisible by the base-14 number 16.

See The Solution Submitted by K Sengupta    
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1st attempt | Comment 1 of 3

I made a table in Excel of what each of the numbers 1 to 7 would contribute (mod 20) depending on what position that number would be in.  For any digit in position 3 through 7 (1 being the one's digit, and 7 being the 14^6 digit), the modulo is a multiple of 4.  That means digits 1 and 2 must have a modulo that is, for starters, a multiple of 4.  For this to happen, digit 2 (the 14's column) must be 1,3,5, or 7 and digit 1 must be 2 or 6.  For example, the base 14 number 0000052 is 72 which is 12 modulo 20.

So far, without an exhaustive search, I haven't found any combinations of the remaining digits that lead to 0 mod 20.

I am guessing that either none exists or the number is very small.  But this is just based on an inspection of my Excel table.

  Posted by Larry on 2010-09-12 13:43:18
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