Exercise · ZW-MODULAR-INVERSE-001 · ★☆☆ Basic
What is the necessary and sufficient condition for \(a\) to have an inverse modulo \(m\)?
Choose one option.
Show answer
- D. \(\gcd(a,m)=1\)
Explanation
A modular inverse exists exactly when \(a\) and \(m\) are coprime. The other options are neither necessary nor sufficient for existence.
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-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.
- ZW-MODULAR-INVERSE-009 If b is an inverse of a modulo m and c≡ apmodm, what holds?