PEI Math 521B study plan

State and prove the division algorithm, Euclidean algorithm, and Bézout's identity from first principles

Solve 15 linear Diophantine equations of the form ax + by = c, including cases with no solution

Classify integers as prime or composite using trial division and the Sieve of Eratosthenes up to 200

Derive the fundamental theorem of arithmetic and write a rigorous proof of unique factorization

Compute gcd and lcm via prime factorization and via the Euclidean algorithm for 20 integer pairs

Prove divisibility properties including transitivity and linear combinations, and apply them to contest-style problems

Solve 12 modular arithmetic problems involving addition, multiplication, and inverses modulo n

Translate 5 word problems into congruence equations and solve them completely

Factor 20 polynomials over the integers and rationals, including grouping and substitution methods

State and apply the Factor Theorem, Remainder Theorem, and Rational Root Theorem to 10 polynomial problems

Solve 15 quadratic and higher-degree equations, including those requiring completing the square and the quadratic formula

Prove 5 algebraic identities using induction or direct manipulation

Solve 12 systems of linear equations using substitution, elimination, and matrix methods

Apply the binomial theorem to expand 8 expressions and find specific terms

State and use the triangle congruence criteria (SSS, SAS, ASA, AAS, HL) to prove 8 geometric statements

Compute areas and perimeters of 10 composite figures by decomposing them into standard shapes

Prove the Pythagorean theorem using at least two distinct methods and apply it to 10 problems

Solve 10 problems involving similar triangles, including scale factors and proportional segments

Derive and apply the formulas for the area of a circle, sector, and segment to 8 problems

Use coordinate geometry to prove 5 properties of triangles and quadrilaterals

Apply the fundamental counting principle, permutations, and combinations to solve 15 counting problems

Distinguish between permutations and combinations and justify the choice in 10 word problems

Use the pigeonhole principle to prove 5 existence statements

Compute probabilities for 10 events using counting techniques and complementary counting

Solve 8 combinatorial problems using inclusion-exclusion and complementary counting

Write 5 complete proofs by contradiction and 5 by contrapositive, clearly identifying the assumption and conclusion

Analyze 6 flawed proofs, identify the logical error, and rewrite them correctly

Construct a counterexample to disprove 5 false statements

Complete two full-length timed practice exams under test conditions and score each

Review all errors from practice exams and categorize them by domain and error type

Re-solve 20 previously missed problems without notes to confirm mastery

Summarize key theorems and formulas from all five domains into a single reference sheet