Exercise · ZW-LINEAR-CONGRUENCE-010 · ★★★ Advanced
Find all solutions of \(6x \equiv 9 \pmod{15}\) modulo 15.
Choose one option.
Show answer
- B. {4, 9, 14}
Explanation
Since \(\gcd(6,15)=3\) divides 9, there are three solutions. Dividing by 3 gives \(2x\equiv 3\pmod{5}\), whose inverse of 2 is 3, yielding \(x\equiv 4\pmod 5\), so the solutions are 4, 9 and 14.
Review the concept: Linear congruence
A linear congruence is a first-degree congruence in one unknown \(x\), with integer coefficients \(a\), \(b\) and modulus \(m\); its solutions are the integers \(x\) that satisfy it.
\[ax \equiv b \pmod{m}\]
More practice problems: linear congruence
- ZW-LINEAR-CONGRUENCE-001 What is the modular inverse of 3 modulo 7?
- ZW-LINEAR-CONGRUENCE-002 What is the greatest common divisor of 4 and 6?
- ZW-LINEAR-CONGRUENCE-003 How many solutions modulo 10 does the congruence 3x ≡ 1 pmod10 have?
- ZW-LINEAR-CONGRUENCE-004 Does the congruence 2x ≡ 1 pmod4 have a solution?
- ZW-LINEAR-CONGRUENCE-005 What is the value of x in x ≡ 3 pmod5 that lies between 0 and 4?
- ZW-LINEAR-CONGRUENCE-006 Solve 3x ≡ 2 pmod7. What is the solution modulo 7?
- ZW-LINEAR-CONGRUENCE-007 Solve 5x ≡ 4 pmod9. What is the solution modulo 9?
- ZW-LINEAR-CONGRUENCE-008 Solve 2x ≡ 4 pmod6. How many solutions modulo 6 does it have?