Study Guide — Divisibility and congruences

Number theory

Undergraduate study guide for Divisibility and congruences.

What this standard means

• Divisibility, GCD, and the Euclidean algorithm
• Bézout's identity and linear Diophantine equations
• Prime numbers; fundamental theorem of arithmetic
• Congruences and modular arithmetic
• Fermat's little theorem and Euler's theorem

_See printable PDF for diagram._

How to use the 20 practice sets

| Sets | When to use | | --- | --- | | 1–5 | Intro — explore together, short written items | | 6–10 | Core skills — diagrams and written practice | | 11–15 | Mixed review — explain thinking | | 16–20 | Stretch — word problems and mastery tasks |

Pacing: 10–15 minutes per session.

How to practice

1. Use trees or tables for counting 2. State the proof method up front 3. Check small cases before generalizing

_See printable PDF for diagram._

Common mistakes

• Double counting
• Missing base case in induction

Review and practice tests

1. Start Review 1/10 when sets 1–3 feel comfortable. 2. Move up one review level with little help. 3. Use Practice Test 4/10–6/10 for mid-standard checks. 4. Practice Test 10/10 is the mastery bar for Divisibility and congruences.

• [ ] Applies counting principles correctly
• [ ] Writes structured proofs
• [ ] Models problems with graphs or recurrences

Materials for this standard

• Practice Problems — 20 printable sets
• Review — 10 difficulty levels
• Practice Test — 10 difficulty levels
• Answer key — for parents and tutors