Exercise · ZW-MODULAR-INVERSE-009 · ★★☆ Intermediate
If \(b\) is an inverse of \(a\) modulo \(m\) and \(c\equiv a\pmod{m}\), what holds?
Choose one option.
Show answer
- D. \(b\) is also an inverse of \(c\) modulo \(m\)
Explanation
Since \(c\equiv a\pmod{m}\), we get \(bc\equiv ba\equiv 1\pmod{m}\), so \(b\) is also an inverse of \(c\). The other options fail: \(c\) is coprime to \(m\) and \(b+m\equiv b\).
Review the concept: Modular inverse
The modular inverse of an integer \(a\) modulo \(m\) is an integer \(b\) whose product with \(a\) is congruent to 1 modulo \(m\).
\[ab \equiv 1 \pmod{m}\]
More practice problems: modular inverse
- ZW-MODULAR-INVERSE-001 What is the necessary and sufficient condition for a to have an inverse modulo m?
- ZW-MODULAR-INVERSE-002 Does 2 have an inverse modulo 4?
- ZW-MODULAR-INVERSE-003 State the uniqueness property of the modular inverse modulo m.
- ZW-MODULAR-INVERSE-004 Which integer b satisfies 3b≡ 1pmod5?
- ZW-MODULAR-INVERSE-005 Which of these congruences has a solution in b?
- ZW-MODULAR-INVERSE-006 Find the inverse of 5 modulo 11.
- ZW-MODULAR-INVERSE-007 Which pair of numbers are inverses of each other modulo 10?
- ZW-MODULAR-INVERSE-008 Solve 3x≡ 2pmod7 using the inverse of 3.