Perform prime factorization of $30$ , $40$ and $50$ by factor tree method. Hence find the highest common factor of $30$ , $40$ and $50$.
A. $5$
B. $20$
C. $40$
D. $10$

Hint: In this problem we need to find the highest common factor of the given number by calculating the prime factors of the given numbers. So we will consider each number individually and find the prime factors of the number by using the factor tree method. After having the prime factors of all the numbers we will calculate the highest common factor of the numbers by multiplying the common prime factors in all the given numbers.

Complete step by step answer:
Given numbers are $30$ , $40$ and $50$.
Considering the number $30$. The prime factors of the number $30$ can be calculated by using factor tree method as
$\begin{align}
  & 2\left| \!{\underline {\,
  30 \,}} \right. \\
 & 3\left| \!{\underline {\,
  15 \,}} \right. \\
 & 5\left| \!{\underline {\,
  5 \,}} \right. \\
 & \left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}$
So we can write the prime factorization of the number $30$ from above tree is given by
$30=2\times 3\times 5$
Considering the number $40$. The prime factors of the number $40$ can be calculated by using factor tree method as
$\begin{align}
  & 2\left| \!{\underline {\,
  40 \,}} \right. \\
 & 2\left| \!{\underline {\,
  20 \,}} \right. \\
 & 2\left| \!{\underline {\,
  10 \,}} \right. \\
 & 5\left| \!{\underline {\,
  5 \,}} \right. \\
 & \left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}$
So we can write the prime factorization of the number $40$ from above tree is given by
$40=2\times 2\times 2\times 5$
Considering the number $50$. The prime factors of the number $50$ can be calculated by using factor tree method as
$\begin{align}
  & 2\left| \!{\underline {\,
  50 \,}} \right. \\
 & 5\left| \!{\underline {\,
  25 \,}} \right. \\
 & 5\left| \!{\underline {\,
  5 \,}} \right. \\
 & \left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}$
So we can write the prime factorization of the number $50$ from above tree is given by
$50=2\times 5\times 5$
So we have the prime factorization of the all numbers as
$30=2\times 3\times 5$, $40=2\times 2\times 2\times 5$ and $50=2\times 5\times 5$.
We can observe that the factors $2$ and $5$ are common in all the terms. So the highest common factor of all the numbers is given by
$\begin{align}
  & \text{H}\text{.C}\text{.F}=2\times 5 \\
 & \Rightarrow \text{H}\text{.C}\text{.F}=10 \\
\end{align}$

Hence the highest common factor of the numbers $30$ , $40$ and $50$ is $10$ .

So, the correct answer is “Option D”.

Note: In this problem we have only asked to calculate the highest common factor. If they have asked to calculate the Least Common Multiple, then we can write the least common multiple from the prime factorization as
$\begin{align}
  & \text{L}\text{.C}\text{.M}={{2}^{3}}\times {{3}^{1}}\times {{5}^{2}} \\
 & \Rightarrow \text{L}\text{.C}\text{.M}=600 \\
\end{align}$
Hence the Least Common Multiple of the numbers $30$ , $40$ and $50$ is $600$.