imc

Question

In base 10, the number 2013 ends in the digit 3. In base 9, on the other hand, the same number is written as (2676)_9 and ends in the digit 6. For how many positive integers b does the base-b-representation of 2013 end in the digit 3?

Accepted Answer

We want the integers b such that 2013 3b b is a factor of 2010. Since 2010=2 3 5 67, it has (1+1)(1+1)(1+1)(1+1)=16 factors. Since b cannot equal 1, 2, or 3, as these cannot have the digit 3 in their base representations, our answer is 16-3=13