Prime Numbers and Prime Factorisation with Factor Trees

Q: Why is 1 not a prime number?

A prime has exactly two different factors, and 1 has only one. If 1 counted as prime, factorisations would stop being unique, since you could add as many × 1s as you liked.

Q: Is 2 the only even prime?

Yes. Every other even number is divisible by 2, so it has at least three factors: 1, 2 and itself.

What makes a number prime, why 1 is not prime, how to break any number into primes with a factor tree, and why every number has exactly one prime factorisation.

3 min read Supports 3 lessons

Some numbers can be built by multiplying smaller whole numbers together: $12 = 3 \times 4$ . Others cannot. The numbers that cannot be broken down any further are the primes: $2, 3, 5, 7, 11, 13, 17, 19, 23, \dots$

A prime has exactly two factors: $1$ and itself. A number with more than two factors is composite.

Key idea

$1$ is neither prime nor composite. It has only one factor, so it does not meet the definition. Leaving it out keeps the theorem below true.

Primes as building blocks

Every composite number can be split into factors, and those factors split again, until nothing is left but primes. Drawing this splitting as branches gives a factor tree.

Each branch ends when it reaches a prime. Collect the leaves: $360 = 2^3 \cdot 3^2 \cdot 5$ .

You can start a factor tree with any split you like. $360 = 36 \times 10$ and $360 = 2 \times 180$ lead to different-looking trees, but the leaves at the bottom are always the same primes.

One number, one factorisation

Key idea

The Fundamental Theorem of Arithmetic: every whole number greater than $1$ can be written as a product of primes in exactly one way, apart from the order of the factors.

This is why primes are often called the atoms of arithmetic. Just as a molecule has one chemical formula, a number has one prime factorisation.

Finding a prime factorisation

A reliable method is to divide by the smallest prime that works, again and again, until you reach $1$ .

$84$
Start with 84. It is even, so divide by 2.

1/5

Is it prime?

To test whether a number is prime, try dividing by the primes $2, 3, 5, 7, \dots$ up to its square root. If none of them divide it, it is prime.

Worked example

Is $91$ prime?

1
$\sqrt{91}$ is a little under $10$ , so test $2, 3, 5$ and $7$ .

Tip

Quick tests: even numbers are divisible by $2$ ; if the digits add to a multiple of $3$ , so is the number; numbers ending in $0$ or $5$ are divisible by $5$ .

Where you will use it

Prime factorisations make the greatest common factor and least common multiple quick to find, help simplify fractions and square roots, and show at a glance whether one number divides another.

Common questions

Why is 1 not a prime number?: A prime has exactly two different factors, and $1$ has only one. If $1$ counted as prime, factorisations would stop being unique, since you could add as many $\times 1$ s as you liked.
Is 2 the only even prime?: Yes. Every other even number is divisible by $2$ , so it has at least three factors: $1$ , $2$ and itself.

Practise prime factorisation

The Primes & Prime Factorization lesson has worked examples and unlimited practice that adapts to you.

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Common questions

Why is 1 not a prime number?

A prime has exactly two different factors, and $1$ has only one. If $1$ counted as prime, factorisations would stop being unique, since you could add as many $\times 1$ s as you liked.

Is 2 the only even prime?

Yes. Every other even number is divisible by $2$ , so it has at least three factors: $1$ , $2$ and itself.