# 1982 AHSME Problems/Problem 26

## Problem 26

If the base representation of a perfect square is , where , then equals

## A Solution

A perfect square will be where .

Notice that .

Now in base 8 is . It being a perfect square means . That means that c can only be 1 so the answer is 1 = .

## Partial and Wrong Solution

From the definition of bases we have , and

If , then , which makes

If , then , which clearly can only have the solution , for . This makes , which doesn't have 4 digits in base 8

If , then , which clearly can only have the solution , for . is greater than , and thus, this solution is invalid.

If , then , which clearly has no solutions for .

Similarly, yields no solutions

If , then , which clearly can only have the solution , for . This makes , which doesn't have 4 digits in base 8.

If , then , which clearly can only have the solution , for . This makes , which doesn't have 4 digits in base 8