Factor definition
A factor is a number that divides exactly into another number with no remainder. For example, 3 is a factor of 12 because 12 divided by 3 equals 4 exactly. The number 5 is not a factor of 12 because 12 divided by 5 leaves a remainder. Factors are closely connected to multiplication and division because they show which numbers can be multiplied together to make another number.
Factor pairs are two numbers that multiply together to make a given number. For 24, the factor pairs are 1 and 24, 2 and 12, 3 and 8, and 4 and 6. This means the factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24. Listing factor pairs in order is a good method because it helps students avoid missing any factors.
Factors are used in number, fractions, ratio, algebra, factorisation, prime numbers, multiples, divisibility, highest common factor and problem solving. At GCSE level, students use factors to simplify fractions, find common factors, factorise expressions, solve equations and understand prime factor decomposition. A-Level maths also uses factors in polynomial algebra, roots of equations and simplifying expressions.
A common factor is a factor shared by two or more numbers. For example, the factors of 18 are 1, 2, 3, 6, 9 and 18. The factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24. The common factors are 1, 2, 3 and 6. The highest common factor, often called the HCF, is 6. This is useful when simplifying fractions and sharing quantities into equal groups.
Prime factors are factors that are also prime numbers. For example, 60 can be written as 2 times 2 times 3 times 5. These are its prime factors. Prime factorisation helps students find highest common factors, lowest common multiples and solve number problems systematically. It is also useful for checking whether a number is a square number or cube number.
Factors also appear in algebra. In the expression 3x + 6, both terms have a common factor of 3. This means the expression can be written as 3(x + 2). In this form, 3 and x + 2 are factors of the expression. Recognising algebraic factors is important for factorising, simplifying fractions, solving quadratic equations and working with expressions in different forms.
Students sometimes confuse factors with multiples. A factor divides into a number. A multiple is the result of multiplying a number. For example, 4 is a factor of 20 because it divides into 20 exactly. But 20 is a multiple of 4 because 4 times 5 equals 20. Remembering this difference is important in exam questions about HCF and LCM.
Related ideas include multiples, prime numbers, divisibility, factor pairs, common factors, highest common factor, prime factorisation, factorising and simplifying fractions. A clear definition of factor helps students understand how numbers and expressions can be broken into useful parts. This makes many maths topics more organised, especially when solving problems that involve sharing, grouping, simplifying or rewriting expressions.