Highest common factors and lowest common multiples using prime factors
In some questions the The highest common factor (HCF) of two numbers is the largest number which will divide exactly into both of them, for example, the highest common factor of 24 and 36 is 12. (HCF) or The smallest positive number that is a multiple of two or more numbers. (LCM) of two large numbers may need to be found.
It would take a long time to write out all the factors and multiples of 24 and 180 and compare the lists and it would be easy to make a mistake.
A more efficient method is to use prime factors.
Using prime factors to find the HCF and LCM
Numbers can be broken down into prime factors using prime factor trees. When the prime factors of two numbers are known, they can be compared to calculate HCFs and LCMs. This can be a more efficient method than listing the factors and multiples of large numbers.
Example
Find the highest common factor and lowest common multiple of 24 and 180.
Start by listing the prime factors of 24 and 180.
The product of prime factors for 24 is: \(2 \times 2 \times 2 \times 3\)
The product of prime factors for 180 is: \(2 \times 2 \times 3 \times 3 \times 5\)
To find the HCF, find any prime factors that are in common between the lists. Each list contains two 2s and one 3, so use these for the HCF.
HCF = \(2 \times 2 \times 3 = 12\)
Cross any numbers used so far off the lists.
The product of prime factors for 24 is: \(\cancel2\times\cancel2\times2\times\cancel3\)
The product of prime factors for 180 is: \(\cancel2\times\cancel2\times\cancel3\times3\times5\)
To find the LCM, multiply the HCF by all the numbers in the lists that have not yet been used.
LCM = \(12 \times 2 \times 3 \times 5 = 360\)