Identify primes; write a number as a product of its prime factors using index notation; and find the highest common factor (HCF) and lowest common multiple (LCM) of two numbers, including from prime factorisation.

What this dot point is asking

The Eduqas number content requires you to identify prime numbers, write any integer as a product of its prime factors in index form, and use that factorisation to find the highest common factor (HCF) and lowest common multiple (LCM) of two numbers. Prime factorisation is the engine behind HCF and LCM, behind simplifying fractions and surds, and behind many problem-solving questions about repeating events and equal groupings. It appears at both tiers, with the HCF and LCM from prime factors being a reliable mid-tariff question.

Primes and prime factorisation

A prime number has exactly two distinct factors.

The fundamental theorem of arithmetic says every integer greater than $1$ is either prime or can be written as a product of primes in exactly one way (ignoring order). A factor tree finds that product: split the number into any factor pair, then keep splitting composite branches until every branch ends in a prime. Collecting repeated primes into index form is the expected final step, so $72 = 2 \times 2 \times 2 \times 3 \times 3 = 2^3 \times 3^2$.

Factors and multiples

A factor of a number divides it exactly with no remainder, so the factors of $12$ are $1, 2, 3, 4, 6, 12$. Factors come in pairs that multiply to the number ($2 \times 6$, $3 \times 4$), which is why listing them in pairs guarantees none are missed. A multiple is the result of multiplying the number by an integer, so the multiples of $12$ are $12, 24, 36, 48, \ldots$. The highest common factor of two numbers is the largest factor they share, and the lowest common multiple is the smallest multiple they share. For small numbers you can list and compare, but prime factorisation is faster and avoids missed factors once the numbers grow. Once a number is in index form, the total number of factors is found by adding one to each index and multiplying: $360 = 2^3 \times 3^2 \times 5^1$ has $(3+1)(2+1)(1+1) = 24$ factors, a neat Higher-tier reasoning result.

Finding the HCF and LCM

Once both numbers are in prime-factor form, the HCF and LCM follow rules.

For $24 = 2^3 \times 3$ and $36 = 2^2 \times 3^2$: the HCF takes the lowest powers of the shared primes, $2^2 \times 3 = 12$; the LCM takes the highest powers, $2^3 \times 3^2 = 72$. Checking, $12 \times 72 = 864 = 24 \times 36$, which confirms both. HCF problems ask for the largest equal grouping (the biggest box, the most identical bunches); LCM problems ask when repeating events coincide (lights flashing, buses arriving).

Why this matters

Prime factorisation is reused constantly. Simplifying a fraction means cancelling shared prime factors; simplifying a surd means pulling out a square factor; finding a common denominator uses the LCM of the denominators. Because Eduqas weights AO2 and AO3 (reasoning and problem solving) at half the marks, the worded HCF and LCM questions, deciding which one a real-world situation needs, test understanding rather than mechanical calculation, so reading the context carefully is the key skill.