Factors, Multiples and Prime

Two words sit at the heart of this topic, and they pull in opposite directions.

A factor of a number divides into it exactly, with nothing left over. The factors of 12 are $1, 2, 3, 4, 6, 12$ — each one divides 12 with no remainder.

A multiple of a number is what you get by multiplying it by a whole number. The multiples of 12 are $12, 24, 36, 48, \dots$ — the answers in its times table.

A neat way to keep them apart: factors are smaller than or equal to the number and there are only a few of them, while multiples are equal to or bigger than the number and the list goes on forever.

To list factors without missing any, search in pairs. For 12: $1 \times 12$, $2 \times 6$, $3 \times 4$ — work upwards until the pair meets in the middle.

Once you can list factors and multiples for one number, you can compare two numbers and look for the ones they have in common.

A common factor is a number that is a factor of both numbers. The factors of 8 are $1, 2, 4, 8$ and the factors of 12 are $1, 2, 3, 4, 6, 12$. The ones in both lists — $1, 2, 4$ — are the common factors.

A common multiple works the same way the other direction. The multiples of 4 are $4, 8, 12, 16, 20, 24, \dots$ and the multiples of 6 are $6, 12, 18, 24, \dots$ The shared ones — $12, 24, \dots$ — are the common multiples.

Spotting common factors and multiples is the gateway to the HCF and LCM later in this guide, and it is also how you simplify fractions and find common denominators.

A prime number is a whole number greater than 1 that has exactly two factors — 1 and the number itself. Nothing else divides into it.

So 7 is prime because its only factors are 1 and 7. But 9 is not prime, because 3 divides into it as well, giving it three factors: $1, 3, 9$. A number with more than two factors is called a composite number.

Two facts trip people up. First, 1 is not prime — it has only one factor, not two. Second, 2 is prime, and it is the only even prime, because every other even number can also be divided by 2.

The primes up to 20 are $2, 3, 5, 7, 11, 13, 17, 19$. They are worth knowing well.

Here is a powerful idea: every whole number bigger than 1 can be built by multiplying prime numbers together. Writing a number this way is called prime factorisation.

The friendliest method is a factor tree. Split the number into any two factors, then keep splitting until every branch ends on a prime.

The green numbers at the bottom — $2, 2, 3, 3$ — are the prime factors. So $36 = 2 \times 2 \times 3 \times 3$, which is tidier still as $2^2 \times 3^2$.

A wonderful fact: it does not matter how you start the splitting. Begin 36 as $6 \times 6$ or as $2 \times 18$ and you always end on exactly the same set of primes.

Two special values describe how a pair of numbers relate.

The highest common factor (HCF) is the biggest number that divides into both. For 8 and 12, the common factors are $1, 2, 4$, so the HCF is 4.

The lowest common multiple (LCM) is the smallest number that both divide into. For 4 and 6, the common multiples are $12, 24, \dots$, so the LCM is 12.

For small numbers, listing works fine. For larger numbers, prime factorisation is faster. Write both numbers as products of primes, then for the HCF multiply the primes they share, and for the LCM take every prime, at its highest power.

The HCF helps you simplify fractions; the LCM helps you add fractions and solve "when do two events meet again" problems.

These ideas turn up again and again as you move forward:

• Fractions lean on the HCF for simplifying and on the LCM for finding common denominators when adding or subtracting.
• Algebra uses factors when you factorise expressions — pulling out a common factor mirrors finding the HCF of numbers.
• Problem solving often hides an LCM, such as working out when two buses arriving at different intervals next leave together.
• Powers and roots connect here too, since prime factorisation is the clearest way to write a number using index notation.

If a fractions topic later feels shaky, it is often the HCF or LCM underneath that needs a quick refresh. These number-building skills quietly support a huge amount of maths to come.