Number Theory: Primes & Divisibility
Number theory explores the properties of integers, including divisibility rules, prime factorisation, and the relationships between numbers that underpin modern cryptography and computation.
What You Need to Know
Key Concept Diagram
A prime number has exactly two distinct factors: 1 and itself; composite numbers have more than two factors
Every composite number can be expressed uniquely as a product of prime factors — the Fundamental Theorem of Arithmetic
The Highest Common Factor (HCF) is found by multiplying shared prime factors; the Lowest Common Multiple (LCM) uses all prime factors
Divisibility rules provide quick tests: a number is divisible by 9 if the digit sum is divisible by 9; by 3 if the digit sum is divisible by 3; by 6 if divisible by both 2 and 3
Key Vocabulary
Prime number
A whole number greater than 1 that has exactly two factors: 1 and itself
Prime factorisation
Expressing a composite number as a product of its prime factors (e.g., 60 = 2² × 3 × 5)
HCF (Highest Common Factor)
The largest integer that divides evenly into two or more numbers
LCM (Lowest Common Multiple)
The smallest positive integer that is a multiple of two or more given numbers
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
What is the prime factorisation of 180?
Question 2
Find the HCF of 48 and 72.
Question 3
Is 561 divisible by 3? How do you know?
Key Concepts Summary
- ●A prime number has exactly two distinct factors: 1 and itself; composite numbers have more than two factors
- ●Every composite number can be expressed uniquely as a product of prime factors — the Fundamental Theorem of Arithmetic
- ●The Highest Common Factor (HCF) is found by multiplying shared prime factors; the Lowest Common Multiple (LCM) uses all prime factors
- ●Divisibility rules provide quick tests: a number is divisible by 9 if the digit sum is divisible by 9; by 3 if the digit sum is divisible by 3; by 6 if divisible by both 2 and 3