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Prime numbers

A prime number has exactly two factors, itself and one.

The first ten prime numbers are \({2}\), \({3}\), \({5}\), \({7}\), \({11}\), \({13}\), \({17}\), \({19}\), \({23}\) and \({29}\).

Key points

• The only even prime number is \({2}\). All other even numbers are divisible by \({2}\).

• The number \({1}\) is not prime because a prime number has exactly two factors and \({1}\) only has one factor.

Factors

The factors of a number are the numbers that divide into it exactly.

The factors of \({12}\) are \({1}\), \({2}\), \({3}\), \({4}\), \({6}\) and \({12}\).

The factors of \({10}\) are \({1}\), \({2}\), \({5}\) and \({10}\).

The factors of \({11}\) are \({1}\) and \({11}\) (so \({11}\) is a prime number).

Multiples

The multiples of a number are the numbers that belong to that number's times table.

The multiples of \({3}\) are \({3}\), \({6}\), \({9}\), \({12}\), \({15}\) \({…}\)

The multiples of \({11}\) are \({11}\), \({22}\), \({33}\), \({44}\), \({55}\) \({…}\)

Factors are often given as pairs of numbers, which multiply together to give the original number.

These are called factor pairs.

For example, the factor pairs of \({12}\) are:

• \({1}\) and \({12}\)
• \({2}\) and \({6}\)
• \({3}\) and \({4}\)

A square number will have one factor pair consisting of one factor multiplied by itself. This factor is called the square root of the given number.

The factor pairs of \({16}\) are:

• \({1}\) and \({16}\)
• \({2}\) and \({8}\)
• \({4}\) and \({4}\)

It is often useful to write a number as the product of its prime factors.

This can be done by listing the factor pairs as successive branches in a factor tree.

The branches continue to expand until all the factors are prime numbers.

The final answer is the product of the prime numbers displayed at the end of these branches.

Note that the number \({1}\) is not used within a factor tree.

The factors of a number are any numbers that divide into it exactly.

The factors of \({12}\) are \({1}\), \({2}\), \({3}\), \({4}\), \({6}\) and \({12}\)

The factors of \({18}\) are \({1}\), \({2}\), \({3}\), \({6}\), \({9}\) and \({18}\)

\({1}\), \({2}\), \({3}\) and \({6}\) are factors of both \({12}\) and \({18}\).

They are known as the common factors of \({12}\) and \({18}\).

So the highest common factor (HCF) of \({12}\) and \({18}\) is \({6}\).

The multiples of a number are all the numbers that it will divide into.

The multiples of \({4}\) are \({4}\), \({8}\), \({12}\), \({16}\), \({20}\), \({24}\), \({28}\), \({32}\), \({36}\), \({40}\), \({44}\) \({…}\)

The multiples of \({6}\) are \({6}\), \({12}\), \({18}\), \({24}\), \({30}\), \({36}\), \({42}\), \({48}\), \({54}\), \({60}\), \({66}\) \({…}\)

\({12}\), \({24}\), and \({36}\) are multiples of both \({4}\) and \({6}\) and are known as the common multiples of \({4}\) and \({6}\).

The lowest number that is a multiple of \({4}\) and \({6}\) is \({12}\).

So the lowest common multiple (LCM) of \({4}\) and \({6}\) is \({12}\).