Exercise · ZW-EULER-S-THEOREM-006 · ★★☆ Intermediate
Compute \(11^{203} \bmod 12\).
Choose one option.
Show answer
- A. 11
Explanation
Since \(11\equiv -1\pmod{12}\), an odd power gives \(-1\), that is 11. The option giving 1 assumes the exponent vanishes, which does not happen with 203.
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-005 By Euler's theorem, which exponent of 5 modulo 8 gives remainder 1?
- 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.