Number Theory Basics Practice & Key

Master integer properties and introductory proof techniques with this Geometry and Number Theory worksheet! Covers divisibility tests, prime factorisation, HCF and LCM calculations, modular arithmetic, basic algebraic proofs, proof by contradiction, and answer key.

What is Included:

Section A: Concept Check: Evaluates foundational definitions of prime and composite numbers, HCF definitions, modular congruence notation, listing primes under 30, and explaining why 1 is excluded from prime numbers to preserve unique factorisation.

Section B: Divisibility and Primes: Focuses on applying divisibility rules for 3, 4, and 9, testing integers for primality up to the square root of n, and explaining divisor upper bounds.

Section C: Prime Factorisation, HCF and LCM: Guides students through writing numbers as products of prime factors, computing highest common factors and lowest common multiples, verifying the HCF-LCM product identity, and solving periodic word problems.

Section D: Modular Arithmetic: Covers evaluating mod operations, checking modular congruence statements, calculating large exponent remainders, and finding terminal digits using cyclic patterns.

Section E: Number Properties and Proof: Focuses on constructing direct algebraic proofs for general integer statements, including sums of even numbers, products and difference of squares for consecutive integers, and divisibility properties.

Section F: Challenge: Features classical mathematical proofs, including proving the irrationality of the square root of 2 by contradiction and proving the infinitude of primes using Euclid’s argument.

Teacher Benefits:

Ready to Print & Go: Zero prep required. Excellent for structured classroom instruction, targeted homework assignments, independent practice, sub plans, or unit assessment preparation.

Complete Solutions Included: Comes with a fully worked answer key detailing conceptual checks, prime factor trees, modular remainder steps, and step-by-step formal algebraic proofs.

Scaffolded & Comprehensive: Progresses logically from basic vocabulary and prime testing to prime factorisation, modular arithmetic, and rigorous mathematical proof methods.