Number Theory
Summary: Number theory explores the properties of integers, including factors, multiples, primes, prime factorisation, highest common factor, lowest common multiple, and divisibility rules. Tags: concepts math y7 number-theory primes Created: 2026-05-30 Last Updated: 2026-05-30
Content
Key Definitions
A multiple of a number is the result of multiplying that number by an integer. For example, the first four multiples of 7 are 7, 14, 21, 28.
A factor of a number is an integer that divides exactly into that number with no remainder. For example, the factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24.
A prime number has exactly two distinct factors: 1 and itself. The number 1 is NOT prime because it has only one factor.
Divisibility Rules
These quick checks determine whether one number divides exactly into another:
| Divisor | Rule |
|---|---|
| 2 | The last digit is even (0, 2, 4, 6, 8) |
| 3 | The sum of the digits is divisible by 3 |
| 4 | The last two digits form a number divisible by 4 |
| 5 | The last digit is 0 or 5 |
| 6 | Divisible by both 2 and 3 |
| 8 | The last three digits form a number divisible by 8 |
| 9 | The sum of the digits is divisible by 9 |
| 10 | The last digit is 0 |
| 11 | The alternating sum of digits is divisible by 11 |
Prime Factorisation
Every integer greater than 1 can be written as a unique product of prime numbers. This is called its prime factorisation.
Two methods to find the prime factorisation:
- Factor tree: Repeatedly split the number into factor pairs until only primes remain
- Repeated division: Divide by the smallest prime factor repeatedly
Highest Common Factor (HCF)
The HCF of two or more numbers is the largest number that divides exactly into all of them.
Method 1 — Listing: List all factors of each number, find the largest common one.
Method 2 — Prime factorisation: Write each number as a product of primes, take the lowest power of each common prime.
Lowest Common Multiple (LCM)
The LCM of two or more numbers is the smallest number that is a multiple of all of them.
Method 1 — Listing: List multiples of each number until you find a common one.
Method 2 — Prime factorisation: Write each number as a product of primes, take the highest power of each prime present.
Properties from Prime Factorisation
A number expressed as 2ᵃ × 3ᵇ × 5ᶜ × … has specific properties:
- The total number of factors = (a+1)(b+1)(c+1)…
- A factor of a number uses prime factors with exponents ≤ those in the factorisation
- A multiple of a number must include all prime factors with at least those exponents
Solving Problems with HCF and LCM
- HCF problems: Splitting items into equal groups, finding the largest possible group size
- LCM problems: Finding when events coincide, synchronisation of repeating patterns