Factors and Multiples

A factor of a whole number is any number that divides into it exactly, with no remainder. The factors of $12$ are $1, 2, 3, 4, 6$ and $12$. Notice they come in pairs that multiply to give 12: $1 \times 12$, $2 \times 6$, $3 \times 4$.

A multiple of a number is what you get when you multiply it by a whole number. The multiples of $4$ are $4, 8, 12, 16, 20, 24, \dots$ — they go on for ever.

The neat trick for finding all the factors of a number is to list pairs going up from $1$. For $36$: $1 \times 36$, $2 \times 18$, $3 \times 12$, $4 \times 9$, $6 \times 6$. Stop when the two numbers in the pair meet (here at $6 \times 6$). The full list is $1, 2, 3, 4, 6, 9, 12, 18, 36$.

A prime number is a whole number greater than 1 with exactly two factors — 1 and itself. The first few primes are $2, 3, 5, 7, 11, 13, 17, 19, 23, 29, \dots$. Note that $2$ is the only even prime, and crucially 1 is not prime because it only has one factor.

A composite number is a whole number greater than 1 that is not prime — it has more than two factors. The Fundamental Theorem of Arithmetic says every composite number can be written as a product of primes in exactly one way (apart from the order). For example $60 = 2 \times 2 \times 3 \times 5 = 2^2 \times 3 \times 5$.

The easiest tool for finding the prime factorisation is a factor tree. Split the number into any factor pair, keep splitting each branch until only primes are left, then read them off the ends.

The starting split does not matter — $60 = 4 \times 15$ or $60 = 2 \times 30$ both end at the same set of primes. Use indices to write repeated primes neatly, as in $60 = 2^2 \times 3 \times 5$.

The Highest Common Factor (HCF) of two numbers is the largest number that divides into both of them. It is sometimes called the Greatest Common Divisor.

For small numbers you can list the factors of each and compare. The factors of $18$ are $1, 2, 3, 6, 9, 18$. The factors of $24$ are $1, 2, 3, 4, 6, 8, 12, 24$. The common factors are $1, 2, 3, 6$ — and the highest is $6$. So $\text{HCF}(18, 24) = 6$.

For larger numbers the reliable method is to prime factorise each number first, then multiply the primes they share — taking the lower power when a prime appears in both.

So $72 = 2^3 \times 3^2$ and $120 = 2^3 \times 3 \times 5$. Both have $2^3$ (lowest power $2^3$) and $3$ (lowest power $3^1$). The HCF is $2^3 \times 3 = 24$.

The Lowest Common Multiple (LCM) of two numbers is the smallest whole number that is a multiple of both.

You can find it by listing multiples until you spot the first one they share. The multiples of $4$ are $4, 8, 12, 16, 20, 24, \dots$. The multiples of $6$ are $6, 12, 18, 24, \dots$. The first they share is $12$, so $\text{LCM}(4, 6) = 12$.

For bigger numbers the prime-factorisation method is much faster. Prime factorise each number, then take every prime that appears, raised to its highest power.

So $72 = 2^3 \times 3^2$ and $120 = 2^3 \times 3 \times 5$. Take $2^3$, $3^2$ and $5$ — the LCM is $2^3 \times 3^2 \times 5 = 360$.

A useful link to remember: $\text{HCF} \times \text{LCM} =$ the product of the two original numbers. For 72 and 120: $24 \times 360 = 8640 = 72 \times 120$.

Lots of real problems are HCF or LCM problems in disguise — the trick is spotting which one.

If two events happen on regular cycles and you want to know when they happen together, that is an LCM question. Two lighthouses flash every $12$ s and $15$ s — when do they next flash together? $\text{LCM}(12, 15) = 60$ s.

If you want to share things into equal groups without leftovers, or cut something into the biggest possible equal pieces, that is an HCF question. You have $24$ red and $36$ blue counters and want identical bags with both colours and nothing left over. The largest number of bags is $\text{HCF}(24, 36) = 12$.

Read the question slowly, decide which one fits, and use prime factorisation if the numbers are large.

Strong factor and multiple work supports almost every later number topic:

• Fractions use the LCM for adding and subtracting, and the HCF for simplifying.
• Ratio problems often need an HCF to write a ratio in its simplest form.
• Algebra factorises expressions by spotting common factors — exactly the same idea as HCF.
• Indices and standard form use the same prime-power language.

Whenever a question mentions "common" — common factor, common multiple, common denominator — your factor work is doing the heavy lifting.