Exercise · ZW-EULER-S-CRITERION-009 · ★★☆ Intermediate
If \(a^{(p-1)/2}\equiv 1\pmod p\) and \(b^{(p-1)/2}\equiv 1\pmod p\), what is \((ab)^{(p-1)/2}\) congruent to modulo \(p\)?
Choose one option.
Show answer
- C. With \(1\)
Explanation
Splitting the product, \((ab)^{(p-1)/2}=a^{(p-1)/2}b^{(p-1)/2}\equiv 1\cdot 1=1\pmod p\).
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
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- ZW-EULER-S-CRITERION-005 If a^(p-1)/2≡ -1pmod p, what can you conclude about a?
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