Congruences
Objective. Solve and manipulate modular equations.
- 1. Learn
- 2. Worked example
- 3. Practice
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Learn: the key idea
A congruence states that two integers leave the same remainder upon division by a modulus, written a ≡ b (mod n) when n divides (a−b). Congruences behave like equations under addition, subtraction, and multiplication, allowing modular equations such as linear congruences ax ≡ b (mod n) to be solved, provided gcd(a,n) divides b. Solutions exist in classes modulo n/gcd(a,n), repeating periodically. For example, solving 3x ≡ 6 (mod 9) gives gcd(3,9) = 3 dividing 6, so solutions are x ≡ 2 (mod 3), i.e. x = 2, 5, 8 modulo 9.
Worked example
Solve the congruence 5x ≡ 3 (mod 11) for x modulo 11.
- 1Find inverse of 5 mod 11: 5×9 = 45 ≡ 1 (mod 11)
- 2So inverse of 5 is 9
- 3Multiply both sides: x ≡ 9×3 (mod 11)
- 4Compute: x ≡ 27 ≡ 5 (mod 11)
Answer: x ≡ 5 (mod 11)
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Start practisingCommon mistake. Students cancel a common factor from both sides of a congruence as in ordinary algebra, turning 4x ≡ 8 (mod 12) into x ≡ 2 (mod 12); correctly dividing by gcd(4,12)=4 gives x ≡ 2 (mod 3), a wider solution set.
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