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2.7

# Highest common factors and lowest common multiples

In the previous step we looked at what is meant by the factors of a number and what is meant by a multiple of a number. For example, 12 is a multiple of 4, 4 is a factor of 12.

In this step we consider common factors and common multiples of two or more numbers. Being able to find the highest common factor helps develop fluency in areas of mathematics. For example: deciding by what number to divide the denominator and numerator of a fraction when expressing the fraction in its simplest form and what number to take as a common factor when factorising an algebraic expression.

## Highest common factors

The highest common factor (HCF) of two or more numbers is the largest number that is a factor of all of the given numbers.

For example: The factors of 8 are 1, 2 ,4,and 8.
The factors of 12 are 1, 2, 3, 4, 6, 12.
The highest common factor of 8 and 12 is 4.
4 is the largest number that is a factor of both 8 and 12.

Being able to find the lowest common multiple is useful when determining which number to use as the denominator when converting two fractions so they can be added or subtracted.

## Lowest common multiples

The lowest common multiple (LCM) of two or more numbers is the smallest number which is a multiple of all of the given numbers.

For example: The multiples of 8 are 8,16, 24,32,40,….
The multiples of 12 are 12, 24, 36, 48,….
The lowest common multiple of 8 and 12 is 24.
24 is the smallest number which is a multiple of both 8 and 12.

These methods of finding the highest common factor and the lowest common multiple can be found in most text books. We will consider an alternate method of finding the HCF and the LCM once we have explored prime numbers.

## Problem worksheet

Now complete questions 5 and 6 from this week’s worksheet.