Difference between revisions of "2021 Fall AMC 12B Problems/Problem 13"
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==Solution 3== | ==Solution 3== | ||
− | + | We have that <math>5^2 \equiv 3 \pmod{11}</math>, so 3 is a quadratic residue mod 11. For quadratic residues, their Legendre symbol which we is the answer from Solution 2 is <math>\boxed{\textbf{(E)}\ 1}</math> | |
==Video Solution (Just 2 min!)== | ==Video Solution (Just 2 min!)== |
Revision as of 18:52, 30 July 2024
Problem
Let What is the value of
Solution
Plugging in , we get Since and we get
~kingofpineapplz ~Ziyao7294 (minor edit)
Solution 2
Eisenstein used such a quotient in his proof of quadratic reciprocity. Let where is an odd prime number and is any integer.
Then is the Legendre symbol . Legendre symbol is calculated using quadratic reciprocity which is . The Legendre symbol
~Lopkiloinm
Solution 3
We have that , so 3 is a quadratic residue mod 11. For quadratic residues, their Legendre symbol which we is the answer from Solution 2 is
Video Solution (Just 2 min!)
~Education, the Study of Everything
See Also
2021 Fall AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 12 |
Followed by Problem 14 |
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 | |
All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.