Congruences
a ≡ b (mod n) when n divides a−b.
Key idea
a ≡ b (mod n) when n divides a−b.
In depth
a ≡ b (mod n) means n | (a − b). Congruence behaves like equality: you can add, subtract and multiply both sides. It is an equivalence relation with n classes.
Worked example
17 ≡ 2 (mod 5) because 5 | 15. And 7 × 8 ≡ 2 × 3 = 6 ≡ 1 (mod 5).
Formula
Real-world uses
- Checking days of the week, clock times and remainders in software and calendars
Common mistake
Dividing both sides without checking for invertibility.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Congruences?
a ≡ b (mod n) when n divides a−b. It is a topic at the advanced level in Discrete Math & Number Theory.
What is the formula for Congruences?
The key formula is a ≡ b (mod n).
Where is Congruences used in real life?
Real-world examples: Checking days of the week, clock times and remainders in software and calendars.
What should I learn before Congruences?
Make sure you are comfortable with Linear Diophantine Equations, the previous topic in Discrete Math & Number Theory.
What is a common mistake with Congruences?
Dividing both sides without checking for invertibility.
How can I practise Congruences?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Congruences?
Move on to Modular Arithmetic next.
Related advanced topics in Discrete Math & Number Theory
- Tautologies and Contradictions – Always true or always false statements.
- Logical Equivalence – Statements with identical truth tables.
- De Morgan’s Laws – Negation swaps AND and OR.
- Converse and Contrapositive – A conditional equals its contrapositive.
- Quantifiers – “For all” and “there exists”.
- Negating Quantified Statements – Flip the quantifier and negate the predicate.