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?

I also couldn’t wait for the computer to find solutions for values of N other than 6, 8 and 10. However, I’ve found them by constructing all possible arrangements of 0s and 3s so that possible multiples can be tested, in increasing order of size for divisibility by 52032.

Maple code below, with tidied up output…

for n from 6 to 16 do
for a to 2^20 do
abn := subs(1 = 3, Reverse(convert(a, base, 2)));
abten := add(n^(j-1)*abn[-j], j = 1 .. nops(abn));
k := abten/52032;
if type(k, integer) then print(n, k, abten, abn); break; end if;
end do; end do;

Base     Multiplier           Multiple of 52032            [in base N]
N
6          753057              39183061824                 [30000033000000]
7          280230408        14580948589056            [3033303333030333]
8          503143              26179536576                 [303033003300]
9          11887539418     618532450997376          [3003033330000303]
10         640625              33333000000                 [33333000000]
11         22094081335     1149599240022720        [303330333003303]
12         556938369        28978617215808            [3300300333000]
13         228368480984   11882468802559488      [303033000303033]
14         643969801797   33507036727101504      [303033333000000]
15         120224169439   6255503984250048        [33330033000033]
16         16228953980     844424933487360          [3000000333300]

Minimum value is 26179536576 (base 10) = 303033003300 (base 8)
Maximum value is 33507036727101504 = 303033333000000 (base 14)

Posted by Harry on 2010-03-17 23:56:57