Exercise · ZW-EULER-S-THEOREM-005 · ★★☆ Intermediate
By Euler's theorem, which exponent of \(5\) modulo \(8\) gives remainder 1?
Choose one option.
Show answer
- B. 4
Explanation
Since \(\varphi(8)=4\), the theorem guarantees \(5^{4}\equiv 1 \pmod 8\). The odd exponents 3, 5 and 7 give \(5\) modulo 8, since \(5^2\equiv 1\).
Review the concept: Euler's theorem
Euler's theorem states that if \(a\) and \(n\) are coprime, then \(a^{\varphi(n)} \equiv 1 \pmod{n}\), generalizing Fermat's little theorem to composite moduli.
\[a^{\varphi(n)} \equiv 1 \pmod{n}, \quad \gcd(a,n)=1\]
More practice problems: euler's theorem
- ZW-EULER-S-THEOREM-001 State Euler's theorem with its hypotheses.
- ZW-EULER-S-THEOREM-002 Compute φ(12).
- ZW-EULER-S-THEOREM-003 Find the remainder of 3^100 modulo 7 using Euler's theorem.
- ZW-EULER-S-THEOREM-004 Which is the value of φ(9)?
- ZW-EULER-S-THEOREM-006 Compute 11^203 bmod 12.
- ZW-EULER-S-THEOREM-007 Use Euler's theorem to find 2^1000 bmod 9.
- ZW-EULER-S-THEOREM-008 Find the last digit of 3^2026 in base 10.
- ZW-EULER-S-THEOREM-009 Compute 5^2025bmod 16.