Question

How do you find the Greatest common factor of 54, 45?

Answer 1

Hint: In the given question, we have been given to calculate the HCF of two numbers. For finding the HCF, we use the relation between LCM, HCF, and the product of two numbers. This gives us the HCF or the GCD of the two numbers.

Formula Used:
We are going to use the relation between LCM, HCF, and the product of two numbers.
\[LCM\left( {a,b} \right) \times HCF\left( {a,b} \right) = a \times b\]

Complete step by step solution:
The given two numbers are \[45\] and \[54\].
To find the HCF, we calculate the LCM.
\[\begin{array}{l}2\left| \!{\overline {\,
 {45,54} \,}} \right. \\3\left| \!{\overline {\,
 {45,27} \,}} \right. \\3\left| \!{\overline {\,
 {15,9} \,}} \right. \\3\left| \!{\overline {\,
 {5,3} \,}} \right. \\5\left| \!{\overline {\,
 {5,1} \,}} \right. \\{\rm{ }}\left| \!{\overline {\,
 {1,1} \,}} \right. \end{array}\]
Hence, the LCM is \[2 \times 3 \times 3 \times 3 \times 5 = 270\]
Now, we divide the product of the numbers by their LCM to find their HCF,
\[HCF = \dfrac{{45 \times{{54}}}}{{{{270}}5}} = 9\]

Hence, the HCF of the two numbers is \[9\].

Additional Information:
HCF of two numbers is the largest number which divides both of the two numbers. It is called the Highest Common Factor. It is also written as GCD; Greatest Common Divisor. While LCM of two numbers or the least common multiple is the smallest number which is divisible by the two numbers.

Note:
In the given question, we had to find the GCD of two numbers. For doing that, we first found the LCM of the two numbers, then multiplied the numbers, and then divided their product by the LCM. This gave us the HCF, HCF, or we can say that we found the GCD.