Division Algorithm
Every integer has a unique quotient and remainder.
Key idea
Every integer has a unique quotient and remainder.
In depth
For integers a and b > 0 there are unique q and r with a = bq + r, 0 ≤ r < b. It formalises division with remainder and underlies the Euclidean algorithm and modular arithmetic.
Worked example
−17 divided by 5: −17 = 5(−4) + 3, so q = −4, r = 3.
Formula
Real-world uses
- Splitting a number of seconds into minutes and seconds in code and timers
Common mistake
Letting the remainder be negative.
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 Division Algorithm?
Every integer has a unique quotient and remainder. It is a topic at the intermediate level in Discrete Math & Number Theory.
What is the formula for Division Algorithm?
The key formula is a = bq + r.
Where is Division Algorithm used in real life?
Real-world examples: Splitting a number of seconds into minutes and seconds in code and timers.
What should I learn before Division Algorithm?
Make sure you are comfortable with Divisibility, the previous topic in Discrete Math & Number Theory.
What is a common mistake with Division Algorithm?
Letting the remainder be negative.
How can I practise Division Algorithm?
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 Division Algorithm?
Move on to Fundamental Theorem of Arithmetic next.
Related intermediate topics in Discrete Math & Number Theory
- Statements and Truth Values – A proposition is true or false, never both.
- Logical Connectives – AND, OR, NOT, implication and biconditional.
- Truth Tables – List all truth assignments to evaluate compound statements.
- Subsets and Power Sets – A set with n elements has 2ⁿ subsets.
- Set Operations – Union, intersection, difference and complement.
- Venn Diagrams – Pictures of sets and their overlaps.