Lesson 3 — Number Theory & Properties

Hands-on · about 10 minutes.

Try these first. They are about how whole numbers are built and how they behave.

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Factors, multiples, and primes

A factor of a number divides it with no remainder. The factors of 12 are 1, 2, 3, 4, 6, 12.
A multiple is what you get by multiplying: the multiples of 4 are 4, 8, 12, 16, … (a multiplication "skip count").
A prime number has exactly two factors: 1 and itself (2, 3, 5, 7, 11, …). A composite number has more than two factors (4, 6, 8, 9, …).

A common trap: 1 is neither prime nor composite. It has only one factor (itself), so it fails the "exactly two factors" test for prime — and it is not composite either. And 2 is the only even prime; every other even number has 2 as an extra factor.

Prime factorization: every number's fingerprint

Every whole number greater than 1 can be written as a product of primes in exactly one way (apart from order). That unique product is its prime factorization.

12 = 2 × 2 × 3 · 18 = 2 × 3 × 3 · 30 = 2 × 3 × 5

A factor tree is the classroom tool: split the number into any two factors, then keep splitting until every branch ends in a prime.

GCF and LCM — and why prime factorization makes them easy

Greatest Common Factor (GCF): the largest number that divides both. GCF of 12 and 18 is 6.
Least Common Multiple (LCM): the smallest number both divide into. LCM of 12 and 18 is 36.

With prime factorizations side by side (12 = 2×2×3, 18 = 2×3×3): the GCF takes the primes they share (one 2 and one 3 → 6); the LCM takes every prime to its highest count (two 2s, two 3s → 2×2×3×3 = 36).

The properties: rules numbers always obey

Property	What it says	Example
Commutative	Order does not matter (+ and ×)	3 + 5 = 5 + 3
Associative	Grouping does not matter (+ and ×)	(2 + 3) + 4 = 2 + (3 + 4)
Distributive	Multiply across a sum	3 × (4 + 5) = 3×4 + 3×5
Identity	The value that leaves a number unchanged	n + 0 = n, n × 1 = n

The identity element is the one that changes nothing: 0 for addition (adding 0 leaves a number alone) and 1 for multiplication (multiplying by 1 leaves it alone). Subtraction and division are not commutative or associative — 5 − 3 is not 3 − 5 — which is exactly why students need these named.

Prime or composite? You decide

Sort each number. Instant feedback either way.

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Quick check

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One sentence summary: primes are the building blocks of every whole number, prime factorization is the tool that unlocks GCF and LCM, and the commutative, associative, distributive, and identity properties name the rules numbers always obey.

Next: Whole Numbers, Integers & Fractions →