Exercise · ZW-MODULAR-CONGRUENCE-005 · ★★☆ Intermediate
Compute \(3^4\bmod5\).
Choose one option.
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- A. 1
Explanation
\(3^2=9\equiv4\) and \(3^4\equiv4^2=16\equiv1\pmod5\). Directly: \(81=16\cdot5+1\).
Review the concept: Modular congruence
Two integers \(a\) and \(b\) are congruent modulo a positive integer \(m\) if \(m\) divides their difference; equivalently, they leave the same remainder when divided by \(m\).
\[a\equiv b\pmod m\iff m\mid(a-b)\]
More practice problems: modular congruence
- ZW-MODULAR-CONGRUENCE-001 Which of the following congruences is true?
- ZW-MODULAR-CONGRUENCE-002 Compute 53bmod8.
- ZW-MODULAR-CONGRUENCE-003 What is -7bmod5?
- ZW-MODULAR-CONGRUENCE-004 Compute (17+25)bmod6.
- ZW-MODULAR-CONGRUENCE-006 Compute 2^10bmod7.
- ZW-MODULAR-CONGRUENCE-007 Solve 3x≡4pmod7 with 0≤ x≤6.
- ZW-MODULAR-CONGRUENCE-008 What is the last digit of 7^2026?
- ZW-MODULAR-CONGRUENCE-009 If a≡ b and c≡ dpmod m, which statement does NOT always hold?