Prime numbers, factors and multiples
Number • Structure and calculation
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Key concepts
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Prime numbers
A prime number has exactly two positive divisors: 1 and itself. Examples of prime numbers include 2, 3, 5 and 7; 2 is the only even prime because any other even number has at least three divisors (1, 2 and itself).
Factors (divisors)
A factor of an integer n is a positive integer that divides n exactly with no remainder. If a × b = n, then both a and b are factors of n. Factors always include 1 and n itself for positive integers greater than 0.
Multiples
A multiple of an integer n is any integer of the form n × k where k is an integer. Multiples form an infinite sequence starting at n, 2n, 3n, and so on. A number is a multiple of n precisely when n divides it exactly.
Common factors and common multiples
A common factor of two or more numbers divides every number in the set exactly. A common multiple of two or more numbers is a number that every number in the set divides exactly. Common factors identify shared divisibility; common multiples identify shared products.
Highest common factor (HCF)
The HCF of two or more integers is the largest integer that divides each of them exactly. HCF reduces by removing shared prime powers. For example, HCF(18, 24) equals 6 because 6 is the largest number that divides both 18 and 24.
Lowest common multiple (LCM)
The LCM of two or more integers is the smallest positive integer that each number divides exactly. LCM grows by combining prime powers to cover all factors present. For example, LCM(6, 8) equals 24 because 24 is the smallest number divisible by both 6 and 8.
Prime factorisation and product notation
Prime factorisation expresses an integer as a product of primes, often written with powers using product notation. For example, 360 = 2^3 × 3^2 × 5. Product notation clarifies repeated prime factors and simplifies comparison of factors across numbers.
Unique factorisation theorem
The unique factorisation theorem (fundamental theorem of arithmetic) states that every integer greater than 1 factors uniquely into primes, up to the order of the factors. Unique prime-power representation enables consistent calculation of HCF and LCM by comparing prime exponents.
Key notes
Important points to keep in mind