Difference between revisions of "1982 AHSME Problems/Problem 26"
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Latest revision as of 15:50, 17 June 2021
Problem 26
If the base representation of a perfect square is , where , then equals
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
See Also
1982 AHSME (Problems • Answer Key • Resources)  
Preceded by Problem 25 
Followed by Problem 27  
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All AHSME Problems and Solutions 
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