A prime number is a number that is only divisible by 1 and itself. Below is a list of the first 15 prime numbers:
\[2,3,5,7,11,13,17,19,23,29,31,37,41,43,47\]
Every number can be written as a product of prime numbers. For example,
\[6=2 \times 3\]
\[8 = 2 \times 2 \times 2 \text{ or } 2^3\]
\[21 = 3\times 7\]
In the above examples,
There may be times when you are asked to find the prime factorisation of a number, but the number is very large. See the video below for an example of finding the prime factorisation of a large number.
The Least Common Multiple of of group of numbers is the smallest number that is divible by all of the numbers in the group.
For example, let's try to find the least common multiple of the numbers \(2,3\) and \(4\) - i.e., we want to find the smallest number that is divisible by \(2,3\) and \(4\).
Let's look at the multiples of each of the numbers and identify the first one that is common to all three:
The smallest number that is in all of the lists is the least common multiple - \(12\).
Writing out the list of multiples and comparing lists is a valid way to find the least common multiple, but this can be difficult if the numbers get large or the lists get long. There is also a method to finding the least common multiple using the prime factorisation of each of the numbers.
Example: Find the least common multiple of \(8,9\) and \(12\).
Let's start by writing the prime factorisations of each:
The only prime numbers present in the prime factorisations are \(2\) and \(3\). The greatest number of times \(2\) appears in the above factorisations is three times (in \(8\)) and the greatest number of times \(3\) appears is twice (in \(9\)). Therefore, the least common multiple is
\[2\times 2\times 2\times 3\times 3 = 72\]
so \(72\) is the least common multiple of \(8,9\) and \(12\). As an exercise, verify this by writing out all the multiples of \(8,9\) and \(12\) and finding the smallest number in each of the lists - i.e., complete the lists below: