Exercise · ZW-MODULAR-CONGRUENCE-008 · ★★☆ Intermediate
What is the last digit of \(7^{2026}\)?
Choose one option.
Show answer
- D. 9
Explanation
The last digit is \(7^{2026}\bmod10\). Powers of 7 give 7, 9, 3, 1 and repeat every 4; since \(2026\equiv2\pmod4\), the digit is 9.
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-009 If a≡ b and c≡ dpmod m, which statement does NOT always hold?