The base ten integer 839 has the representations 1132 and 11324 respectively in base 9 and base 5. Accordingly, 839 has a base 5 representation that begins with its base 9 representation.

For which other base ten integers between 10 and 2014 inclusively do the base 5 representation of the integer begin with its base 9 representation?

Call the base 9 representation a*9^3 + b*9^2 + c*9 + d
and the base 5 representation a*5^4 + b*5^3 + c*5^2 + d*5 + e

Set these equal and simplify to
104a = 44b + 16c + 4d + e
for the base 10 to be 2014 or less a must be 0, 1, or 2
the others can be any of 0 to 4.

I then searched with a program and got the same list as Charlie but didn't have to deal with changing base or truncating so the program was a lot simpler.

Posted by Jer on 2014-05-27 12:45:26