Determine all possible values of a duodecimal (base 12) positive integer N, such that N is obtained by adding the squares of its digits.

Bonus Question:
Determine all possible values of a octodecimal (base 18) positive integer N, such that N is obtained by adding the squares of its digits.

General formula: a(a-1)+b(b-12)+c(c-12^2).... = 0, but since all variables are integers between 0 and 11, c and all following terms must be zero.

Then the solutions are:

• 1;
• 25(12)=29=2^2+5^2;
• A5(12)=125=100+25.

Bonus question: Similarly, the solutions are:

• 1;
• 48(18)=80=8^2+4^2;
• E8(18)=260=14^2+8^2;
• 69(18)=117=6^2+9^2;
• C9(18)=225=12^2+9^2.

Edited on October 17, 2010, 3:22 am

Posted by broll on 2010-10-16 15:58:39