Exercise · ZW-EULER-S-CRITERION-007 · ★★☆ Intermediate
For \(p=13\) and \(a=5\), what is \(5^{6}\bmod 13\), and what does it mean?
Choose one option.
Show answer
- A. \(-1\), so 5 is not a quadratic residue
Explanation
Since \(5^{2}=25\equiv -1\pmod{13}\), we get \(5^{6}\equiv -1\pmod{13}\). This matches the squares modulo 13, which are 1, 3, 4, 9, 10 and 12.
Review the concept: Euler's criterion
Criterion that decides whether an integer \(a\) is a quadratic residue modulo an odd prime \(p\) by means of a modular power.
\[\left(\frac{a}{p}\right) \equiv a^{\frac{p-1}{2}} \pmod{p}\]
More practice problems: euler's criterion
- ZW-EULER-S-CRITERION-001 According to Euler's criterion, what is ((a)/(p)) congruent to modulo p?
- ZW-EULER-S-CRITERION-002 For p=11, which exponent (p-1)/2 does the criterion use?
- ZW-EULER-S-CRITERION-003 For which primes p does Euler's criterion hold as stated in this sheet?
- ZW-EULER-S-CRITERION-004 If a is a quadratic residue modulo p, what is the value of a^(p-1)/2bmod p?
- ZW-EULER-S-CRITERION-005 If a^(p-1)/2≡ -1pmod p, what can you conclude about a?
- ZW-EULER-S-CRITERION-006 Use the criterion to decide whether 3 is a quadratic residue modulo 7. What is the correct conclusi…
- ZW-EULER-S-CRITERION-008 What is 2^(5-1)/2bmod 5, and what does it indicate about 2?
- ZW-EULER-S-CRITERION-009 If a^(p-1)/2≡ 1pmod p and b^(p-1)/2≡ 1pmod p, what is (ab)^(p-1)/2 congruent to modulo p?