Question

Find the HCF and LCM of 32 and 48.

Answer 1

Hint: First of all, we should divide 32 and 48 into prime factors by the method of factorization. After prime factorization, we should have the common factors of 32 and 48. Among the factors of 32 and 48, we have to find the highest common factor of 32 and 48. Now to find LCM, we have to find the multiples of 32 and 48. Among the multiples of 32 and 48, we have to find the least common multiple of 32 and 48.

Complete step-by-step solution -
Before solving the question, we should know the definition of HCF and LCM. HCF is called as the highest common factor. LCM is called least common multiple.
To find HCF of 32 and 48, we have to divide 32 and 48 into factors. We will use the prime factorization method as below,
$\begin{align}
  & 2\left| \!{\underline {\,
  32 \,}} \right. \\
 & 2\left| \!{\underline {\,
  16 \,}} \right. \\
 & 2\left| \!{\underline {\,
  8 \,}} \right. \\
 & 2\left| \!{\underline {\,
  4 \,}} \right. \\
 & 2\left| \!{\underline {\,
  2 \,}} \right. \\
 & 1 \\
\end{align}$
$\begin{align}
  & 2\left| \!{\underline {\,
  48 \,}} \right. \\
 & 2\left| \!{\underline {\,
  24 \,}} \right. \\
 & 2\left| \!{\underline {\,
  12 \,}} \right. \\
 & 2\left| \!{\underline {\,
  6 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3 \,}} \right. \\
 & 1 \\
\end{align}$
From the above, it is clear that $32=2\times 2\times 2\times 2\times 2$ and $48=2\times 2\times 2\times 2\times 3$ .
We get that the highest common factors of 32 and 48 are $2\times 2\times 2\times 2\Rightarrow 16$.
So, the greatest common factor of all common factors of 32 and 48 is equal to 16.
So, the H.C.F of 32 and 48 is equal to 16.
To find LCM of 32 and 48, we have to write down the highest power of each prime factor and then multiply them. We have two prime factors, 2 and 3. 2 has the highest power of 5, and 3 has 1, so we can write the LCM as ${{2}^{5}}\times 3\Rightarrow 96$.
So, we have got the LCM of 32 and 48 is 96.

Note: This question can be solved in another way. To find HCF of 32 and 48, we have to divide 32 and 48 into factors using the prime factorization method,
$\begin{align}
  & 2\left| \!{\underline {\,
  32 \,}} \right. \\
 & 2\left| \!{\underline {\,
  16 \,}} \right. \\
 & 2\left| \!{\underline {\,
  8 \,}} \right. \\
 & 2\left| \!{\underline {\,
  4 \,}} \right. \\
 & 2\left| \!{\underline {\,
  2 \,}} \right. \\
 & 1 \\
\end{align}$
$\begin{align}
  & 2\left| \!{\underline {\,
  48 \,}} \right. \\
 & 2\left| \!{\underline {\,
  24 \,}} \right. \\
 & 2\left| \!{\underline {\,
  12 \,}} \right. \\
 & 2\left| \!{\underline {\,
  6 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3 \,}} \right. \\
 & 1 \\
\end{align}$
From the above, it is clear that $32=2\times 2\times 2\times 2\times 2$ and $48=2\times 2\times 2\times 2\times 3$ .
We get that highest common factors of 32 and 48 are $2\times 2\times 2\times 2\Rightarrow 16$.
So, the H.C.F of 32 and 48 is equal to 16.
We know that the product of two numbers a and b is equal to the product of L.C.M of (a, b) and H.C.F of (a, b).
Product of 32 and 48 \[=32\times \text{48}\Rightarrow \text{1536}\].
Let us assume L.C.M of 32 and 48 is equal to x.
So, we get
\[\Rightarrow 16x=1536\]
Now by using cross multiplication we get,
\[\begin{align}
  & \Rightarrow x=\dfrac{1536}{16} \\
 & \Rightarrow x=96 \\
\end{align}\]
So, the LCM of (32, 48) is equal to 96.