Exercise · ZW-FERMAT-S-LITTLE-THEOREM-008 · ★★☆ Intermediate
What is the value of \(4^{30}\bmod 13\)?
Choose one option.
Show answer
- C. \(1\)
Explanation
Since \(13\) is prime, \(4^{12}\equiv 1\pmod{13}\). Also \(4^{3}=64\equiv -1\pmod{13}\), so \(4^{6}\equiv 1\) and \(4^{30}\equiv 4^{6}\equiv 1\pmod{13}\). The option \(3\) corresponds to \(4^{2}\), an error when reducing the exponent.
Review the concept: Fermat's little theorem
Fermat's little theorem states that if \(p\) is a prime number and \(a\) is an integer not divisible by \(p\), then \(a^{p-1}\equiv 1 \pmod{p}\). It is a basic tool for simplifying modular powers.
\[a^{p-1}\equiv 1 \pmod{p}, \quad p\nmid a\]
More practice problems: fermat's little theorem
- ZW-FERMAT-S-LITTLE-THEOREM-001 By Fermat's little theorem, what is a^p-1 congruent to modulo p when p is prime and pnmid a?
- ZW-FERMAT-S-LITTLE-THEOREM-002 What is the value of 3^6bmod 7?
- ZW-FERMAT-S-LITTLE-THEOREM-003 If p is prime, what condition on a is needed to use a^p-1≡ 1 pmodp?
- ZW-FERMAT-S-LITTLE-THEOREM-004 What is the remainder of 5^12 when divided by 13?
- ZW-FERMAT-S-LITTLE-THEOREM-005 If p=7 and a=2, what value does a^p-1bmod p take?
- ZW-FERMAT-S-LITTLE-THEOREM-006 Compute 2^100bmod 11.
- ZW-FERMAT-S-LITTLE-THEOREM-007 Find the remainder of 3^50 when divided by 7.
- ZW-FERMAT-S-LITTLE-THEOREM-009 Compute 6^17bmod 17.
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