Session 4A
Divisibility definitions
45-75 min - work through lesson notes, practice, and the MCQ check.
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Session tips
- Finish the MCQ before marking complete.
- Check the answer outlines after trying on paper.
- Mark complete to update your certificate progress.
Lesson notes
- Divisibility is a definition disguised as notation. Saying a divides b (a | b) means there exists an integer k such that b = ak. Almost every divisibility proof starts by rewriting using that definition, because it converts a statement about divisibility into an algebra statement about multiples. For example, to show a | (b + c) given a | b and a | c, you write b = ak and c = al, then add them to get b + c = a(k + l), which matches the definition again.
- Divisibility proofs often involve closure properties: multiples of a stay multiples of a under addition and under multiplication by integers. You also learn transitivity: if a|b and b|c then a|c, because c = bq = (ap)q = a(pq). The key writing skill is keeping track of which numbers are integers, because divisibility lives in the integers. Whenever you introduce a k from "a|b," explicitly state k is an integer. That small habit prevents many hidden logical gaps.
Worked examples
- If a | b and a | c then a | (b + c): b = ak, c = al, so b + c = a(k + l).
Practice
- 1If a | b then a | bc for any integer c.
- 2If a | b and b | c then a | c.
- 3If a | b and a | c then a | (mb + nc) for integers m, n.
MCQ
The statement a | b means: