Difference between revisions of "2008 iTest Problems/Problem 83"
Rockmanex3 (talk | contribs) (Solution to Problem 83 -- squares, number theory, and casework) |
Rockmanex3 (talk | contribs) m |
||
Line 13: | Line 13: | ||
&= \frac{n(2n+1)(7n+1)}{6} | &= \frac{n(2n+1)(7n+1)}{6} | ||
\end{align*}</cmath> | \end{align*}</cmath> | ||
− | Thus, <math>(1^ | + | Thus, <math>\left( \sum_{i=1}^n i^2 \right)\left(\sum_{i=n+1}^{2n} i^2 \right) = \frac{n^2 (2n+1)^2 (n+1)(7n+1)}{36}</math>. In order for the expression to be a perfect square, <math>(n+1)(7n+1)</math> must be a perfect square. |
<br> | <br> |
Latest revision as of 21:23, 22 November 2018
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 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50 • 51 • 52 • 53 • 54 • 55 • 56 • 57 • 58 • 59 • 60 • 61 • 62 • 63 • 64 • 65 • 66 • 67 • 68 • 69 • 70 • 71 • 72 • 73 • 74 • 75 • 76 • 77 • 78 • 79 • 80 • 81 • 82 • 83 • 84 • 85 • 86 • 87 • 88 • 89 • 90 • 91 • 92 • 93 • 94 • 95 • 96 • 97 • 98 • 99 • 100 |