Prime Numbers

What you will learn

• Say what a prime number is and why $1$ is not prime.
• Test whether a number is prime.
• Write a number as a product of primes using a factor tree.

Before you start: you should know factors and multiples. See Factors and Multiples.

The idea

A prime number is a whole number greater than $1$ that can only be divided exactly by $1$ and itself. The first few are:

$2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, \ldots$

A whole number greater than $1$ that is not prime is called composite. It can be split into smaller factors: $12 = 3 \times 4$, for instance.

Two facts often surprise people:

• $1$ is not prime. It has only one factor. Excluding it keeps prime factorisations unique.
• $2$ is the only even prime. Every other even number divides by $2$.

Atoms of arithmetic

Every whole number greater than $1$ can be written as a product of primes in exactly one way (ignoring the order). This is the Fundamental Theorem of Arithmetic. Primes are to numbers what atoms are to molecules.

How to test whether a number is prime

You only need to try dividing by primes up to the square root of the number. If $n = a \times b$ and both $a$ and $b$ were bigger than $\sqrt{n}$, their product would be bigger than $n$, which is impossible. So at least one factor must be small.

Worked examples

Example 1: Is 221 prime?

1. $\sqrt{221} \approx 14.9$, so test the primes $2, 3, 5, 7, 11, 13$.
2. $221$ is odd (not divisible by $2$), its digits sum to $5$ (not divisible by $3$) and it doesn’t end in $0$ or $5$.
3. $221 \div 7 = 31.57\ldots$ and $221 \div 11 = 20.09\ldots$, so neither works.
4. $221 \div 13 = 17$ exactly.

Answer: No. $221 = 13 \times 17$, so it is composite.

Example 2: Prime factorisation with a factor tree

Write $360$ as a product of prime factors.

1. Split $360$ into any two factors: $360 = 36 \times 10$.
2. Keep splitting until every branch ends in a prime: $36 = 4 \times 9 = 2 \times 2 \times 3 \times 3$ and $10 = 2 \times 5$.
3. Collect the primes: $2 \times 2 \times 2 \times 3 \times 3 \times 5$.
4. Write with powers: $2^3 \times 3^2 \times 5$.

Answer: $360 = 2^3 \times 3^2 \times 5$

Example 3: Using prime factors to simplify

Prime factorisations make simplifying fractions easy. $\frac{84}{360} = \frac{2^2 \times 3 \times 7}{2^3 \times 3^2 \times 5}$. Cancel the shared $2^2 \times 3$ to get $\frac{7}{2 \times 3 \times 5} = \frac{7}{30}$.

Why it matters

Primes protect your online banking. Encryption schemes such as RSA rely on the fact that multiplying two huge primes is quick, but working backwards to find them from their product is extraordinarily slow, even for powerful computers.

Primes also appear in nature: some cicadas emerge every 13 or 17 years, prime cycles that may help them avoid predators with shorter, regular life cycles. And mathematicians are still hunting for patterns: whether there are infinitely many “twin primes” such as $11$ and $13$ remains unsolved.