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Math help from the Learning Centre

This guide provides useful resources for a wide variety of math topics. It is targeted at students enrolled in a math course or any other Centennial course that requires math knowledge and skills.

Prime Factorisation

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: 

Example

\[6=2 \times 3\]

\[8 = 2 \times 2 \times 2 \text{ or } 2^3\]

\[21 = 3\times 7\]

In the above examples,

  • \(2 \times 3\) is the prime factorisation of \(6\),
  • \(2\times 2\times 2\) is the prime factorisation of \(8\),
  • \(3\times 7\) is the prime factorisation of \(21\).

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.

Least Common Multiple

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:

  • Multiples of \(2\):     \(2,4,6,8,10,\)\(12\),\(14,16,18,20\)
  • Multiples of \(3\):     \(3,6,9,\)\(12\),\(15,18,21\)
  • Multiples of \(4\):     \(4,8,\)\(12\),\(16,20,24\)

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 lowest common multiple using the prime factorisation of each of the numbers. 

  1. Write down the prime factorisation of each of the numbers.
  2. Multiply each factor together, but the amount of times each factor appears in the multiplication is based on the greatest amount of times in appears in any of the prime factorisations.

 

Example

Find the least common multiple of \(8,9\) and \(12\).

Let's start by writing the prime factorisations of each:

  • \(8 = 2\times 2\times 2\)
  • \(9 = 3\times 3\)
  • \(12 = 2\times 2\times 3\)

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:

  • Multiples of \(8\):      \(8,16,24,\) ... 
  • Multiples of \(9\):      \(9,18,27,\) ...
  • Multiples of \(12\):    \(12,24,36,\) ...
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