Exercise · ZW-MODULAR-INVERSE-010 · ★★★ Advanced
Find the inverse of \(7\) modulo \(26\) and use it to solve \(7x\equiv 9\pmod{26}\).
Choose one option.
Show answer
- C. \(x\equiv 5\pmod{26}\)
Explanation
The inverse of \(7\) is \(15\), since \(7\cdot 15=105=4\cdot 26+1\). Then \(x\equiv 15\cdot 9=135\equiv 5\pmod{26}\), and \(7\cdot 5=35\equiv 9\pmod{26}\).
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.