1982 AHSME Problems/Problem 26
Problem 26
If the base representation of a perfect square is , where , then equals
Partial and Wrong Solution
From the definition of bases we have , where is the perfect square.
If , then
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