Chapter 5: Prime Time

Every whole number you will ever write is built from small "atoms" called prime numbers. Just as molecules of water are made of hydrogen and oxygen atoms, every whole number greater than $1$ can be built by multiplying primes. In this chapter you will meet these atoms and learn how to take a number apart into its prime pieces.

We begin with multiples and factors , two ideas that travel together. A multiple of $4$ is anything you get by multiplying $4$ by a whole number: $4, 8, 12, 16, \dots$. A factor of $12$ is any whole number that divides $12$ exactly: $1, 2, 3, 4, 6, 12$. From there we ask: which numbers are factors of only themselves and $1$? Those are the primes: $2, 3, 5, 7, 11, 13, 17, 19, 23, \dots$. Numbers that have more factors are called composite.

We finish with two extraordinarily useful ideas: the prime factorisation of a number, and common factors / common multiples of two numbers. The first lets us tear any number into prime atoms. The second is the secret behind solving everyday problems about packing, sharing, and synchronising things.

What's inside

1. Multiples and factors , the two-way street.
2. Prime and composite numbers , the atoms of arithmetic.
3. Tests of divisibility , quick checks for $2$, $3$, $5$, $9$, $10$.
4. Prime factorisation , breaking a number into prime pieces.
5. Common factors and common multiples (HCF and LCM) , meeting in the middle.

Key results / Formula card

How to read this chapter

Have a piece of squared paper handy. Many of these ideas come alive when you make rectangles of dots , the dimensions of every rectangle that uses $n$ dots are exactly the factor pairs of $n$. Try every exercise; the patterns reveal themselves through doing, not reading.