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.
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