Exercise · ZW-MODULAR-CONGRUENCE-010 · ★★★ Advanced
Which remainders can a perfect square leave when divided by 4?
Choose one option.
Show answer
- A. Only 0 and 1
Explanation
If \(n\) is even, \(n^2\equiv0\); if odd, \(n=2k+1\) and \(n^2=4k(k+1)+1\equiv1\). Never 2 or 3.
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-005 Compute 3^4bmod5.
- 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?