2008 iTest Problems/Problem 83
Problem
Find the greatest natural number such that and is a perfect square.
Solution
Notice that , so Thus, . In order for the expression to be a perfect square, must be a perfect square.
By using the Euclidean Algorithm, . Thus, the GCD of and must be factors of 6. Now, split the factors as different casework. Note that the quadratic residues of 7 are 0, 1, 2, and 4.
- If , then . Let , so . Since 6 is divided out of and , and are relatively prime, so and must be perfect squares. However, since 6 is not a quadratic residue of 7, the GCD of and can not be 6.
- If , then . Let , so . Since 3 is divided out of and , and are relatively prime, so and must be perfect squares. However, since 5 is not a quadratic residue of 7, the GCD of and can not be 3.
- If , then . Let , so . Since 2 is divided out of and , and are relatively prime, so and must be perfect squares. We also know that and do not share a factor of 3, so . That means , so . After trying values of that are one less than a perfect square, we find that the largest value that makes a perfect square is . That means .
- If , then (to avoid common factors that are factors of 6), so . After trying values of that are one less than a perfect square, we find that the largest value that makes a perfect square is (we could also stop searching once gets below 1921).
From the casework, the largest natural number that makes is a perfect square is .
See Also
2008 iTest (Problems) | ||
Preceded by: Problem 82 |
Followed by: Problem 84 | |
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