Make a list of the respective minimum values of a positive base N integer, constituted entirely by threes and zeroes, which is divisible by the base ten number 52032 whenever N is a positive integer with 6 ≤ N ≤ 16.

What are the respective smallest number and the largest number in this list?

Well, assuming Charlie's computer is right (which I haven't checked), there are an infinite number of larger numbers. (which explains why the problem asks only for the maximum of the minima, not the maximum for a given n)

For instance, consider base 6

Assuming 30000033000000 is divisible by 520322, so is

300000330000000 and

3000003300000000 and

30000033000000000 and

300000330000000000 etc.

Also any numbers formed by adding these up is

divisible by 520322

30000033000000 +

3000003300000000, for instance =

3030003333000000, which must be divisible by 52032.

/****************************************/

By the way, the smallest number on Charlie's list

of minima is 26,179,536,576 and the largest

is 39,183,061,824. I assert that this is the answer,

to this puzzle because it makes no sense to compare

the strings of threes and naughts as if they were all

the same base, and then conclude that the longest

string is the largest number. That's like comparing

apples and oranges.

*Edited on ***March 17, 2010, 7:38 pm**