Divisibility
a | b if b = ak for some integer k.
Key idea
a | b if b = ak for some integer k.
In depth
a divides b (a | b) if b = ak for some integer k. Properties: if a | b and a | c then a | (bx + cy), and divisibility is transitive. It is the starting point of number theory.
Worked example
6 | 18 as 18 = 6 × 3. If 7 | 21 and 7 | 35 then 7 | (21 + 35 = 56).
Formula
Real-world uses
- Checking whether a shift of 12 hours wraps back to the same time
- Sharing items equally
Common mistake
Thinking 0 | a is possible; only 0 | 0.
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 Divisibility?
a | b if b = ak for some integer k. It is a topic at the intermediate level in Discrete Math & Number Theory.
What is the formula for Divisibility?
The key formula is a | b.
Where is Divisibility used in real life?
Real-world examples: Checking whether a shift of 12 hours wraps back to the same time; Sharing items equally.
What should I learn before Divisibility?
Make sure you are comfortable with Adjacency Matrices, the previous topic in Discrete Math & Number Theory.
What is a common mistake with Divisibility?
Thinking 0 | a is possible; only 0 | 0.
How can I practise Divisibility?
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 Divisibility?
Move on to Division Algorithm 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.