Summer Problem Solving Marathon Solution #12

Note that since 4² = 16, and since 554 ends in a 4, b must be such that 16 in base b ends in a 4. That means that the remainder when 16 is divided by b is 4. There are then only two possibilities: b = 6 and b = 12.

We then note that in base 10, 24² = 576. Since in base b, 24² = 554, b must be larger than 10. (Try some cases if you don't see why this works.) It thus follows than b = 12. Testing with the actual values, we see that this is correct.

(This can also be solved algebraically, by saying that (2b² + 4)² = 5b³ + 5b² + 4 and solving for b. But the above method is a lot quicker.)