Exercise · ZW-LEGENDRE-SYMBOL-007 · ★★☆ Intermediate
Compute \(\left(\frac{3}{7}\right)\cdot\left(\frac{5}{7}\right)\).
Choose one option.
Show answer
- B. \(1\)
Explanation
The nonzero squares modulo 7 are \(1,2,4\), so \(3\) and \(5\) are non-residues and both symbols equal \(-1\); their product is \(1\) by multiplicativity.
Review the concept: Legendre symbol
The Legendre symbol \(\left(\frac{a}{p}\right)\) for an odd prime \(p\) and an integer \(a\) with \(p \nmid a\) equals \(1\) if \(a\) is a quadratic residue modulo \(p\), and \(-1\) if it is not.
\[\left(\frac{a}{p}\right)\equiv a^{(p-1)/2} \pmod{p}\]
More practice problems: legendre symbol
- ZW-LEGENDRE-SYMBOL-001 What is the value of ((4)/(11))?
- ZW-LEGENDRE-SYMBOL-002 What is the value of ((-1)/(5))?
- ZW-LEGENDRE-SYMBOL-003 If p is an odd prime and p| a, what is ((a)/(p))?
- ZW-LEGENDRE-SYMBOL-004 According to Euler's criterion, what is a^(p-1)/2 congruent to modulo p when pnmid a?
- ZW-LEGENDRE-SYMBOL-005 What is the value of ((2)/(3))?
- ZW-LEGENDRE-SYMBOL-006 Compute ((10)/(13)) given that the nonzero squares modulo 13 are 1,4,9,3,12,10.
- ZW-LEGENDRE-SYMBOL-008 Compute ((2)/(p)) for p=17 using that 6^2≡ 2pmod17.
- ZW-LEGENDRE-SYMBOL-009 Using Euler's criterion, compute 2^3bmod 7 and state which symbol 2 represents modulo 7.