Exercise · ZW-EULER-S-THEOREM-003 · ★★☆ Intermediate
Find the remainder of \(3^{100}\) modulo \(7\) using Euler's theorem.
Choose one option.
Show answer
- D. 4
Explanation
Since \(\varphi(7)=6\) and \(100=6\cdot 16+4\), the remainder is \(3^4=81\equiv 4\). The option giving 1 assumes the exponent reduces to zero modulo 6.
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-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-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.