Determine all possible values of a positive integer N such that the product of the nonzero digits in the base-N representation of 2009 (base ten) is equal to 18 (base ten).
1) By the way, it is clear than the lowest possible base is 4, because bases 2 and 3 do not involve 3 as a digit, and we need a 3 or a 9 to get a product of 18.
2009 base 4 is 33121, so the number base N must be 5 digits or less. It is just a coincidence that base 4 actually works, in addition to giving us a lower bound.
2) Also, the largest base that can possibly work is (2009 minus 18), because 2009 (base 1991) is 1 18. This was guaranteed to work.
3) In general, if M is a positive integer and P the product of its non-zero digits, then base 10 will work.
Base (M - P) will also work if M >= 10.
(M-P) is different from 10 if M = 10 or M > 19.
Edited on June 13, 2010, 11:51 pm