Question

Find the HCF of 144 and 180 by prime factorisation method.

Answer 1

Hint: HCF or Highest Common Factor of a and b can be defined as the largest number that can divide both a and b. To calculate the HCF of two numbers, we should first find the prime factorisation of both numbers. Then, we can write the HCF by multiplying all those prime factors that are present in both of the prime factorisations.

Complete step-by-step solution:
Highest Common Factor (HCF) of a and b is defined as the largest number possible that can perfectly divide both the numbers a and b.
The most common method to find HCF is the Prime Factorisation Method.
In this method, we should first calculate the prime factorisation, then list all the common prime factors and multiply them together to get the highest common factor.
Here, we need to calculate the HCF of 144 and 180.
Prime factorisation of 144:
\[\begin{align}
  & 2\left| \!{\underline {\,
  144 \,}} \right. \\
 & 2\left| \!{\underline {\,
  72 \,}} \right. \\
 & 2\left| \!{\underline {\,
  36 \,}} \right. \\
 & 2\left| \!{\underline {\,
  18 \,}} \right. \\
 & 3\left| \!{\underline {\,
  9 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3 \,}} \right. \\
 & \text{ }\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}\]
$\therefore 144=2\times 2\times 2\times 2\times 3\times 3$
Prime factorisation of 180:
\[\begin{align}
  & 2\left| \!{\underline {\,
  180 \,}} \right. \\
 & 2\left| \!{\underline {\,
  90 \,}} \right. \\
 & 3\left| \!{\underline {\,
  45 \,}} \right. \\
 & 3\left| \!{\underline {\,
  15 \,}} \right. \\
 & 5\left| \!{\underline {\,
  5 \,}} \right. \\
 & \text{ }\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}\]
$\therefore 180=2\times 2\times 3\times 3\times 5$
We can see that the prime factor 2 is repeated at least twice in both the prime factorisations.
Also, we can see that the prime factor 3 is repeated at least twice in both the prime factorisations.
Thus, we can say that the highest common factor will have 2 repeated twice and 3 repeated twice in its prime factorisation.
Hence, we have
$HCF=2\times 2\times 3\times 3$
$\Rightarrow HCF=36$
Hence, the HCF of 144 and 180 is 36.

Note: We must know that the Highest Common Factor (HCF) is also known as Greatest Common Divisor (GCD). We must also note here that the HCF is the product of all the common factors. And, hence there can be no other common factor that is greater than the HCF.