Question

How do you write the least common multiple of $ 10,20, $ and $ 2 $ ?

Answer 1

Hint: In this question, we have to find the least common multiple of 10, 20, and 2. Thus, with the help of the LCM method; we find the least common multiple of the same. First, we find the common factor of 10, 20, and 2, and again and again find the common factor till we get the quotient as 1, 1, and 1. After that, we multiply the common factors to get the least common multiple, which is our required answer.

Complete step by step answer:
According to the question, we have to find the least common multiple.
Numbers given to us are $ \text{10,20, and 2} $
So, let us start solving this problem by finding the common factors of $ \text{10,20, and 2} $ , we get $ \begin{align}
  & \Rightarrow 2\left| \!{\underline {\,
  10,20,2 \,}} \right. \\
 & \Rightarrow \text{ }\left| \!{\underline {\,
  5,10,1 \,}} \right. \\
\end{align} $ -------- (1)
We see that 2 is the common factor of $ \text{10,20, and 2} $ , and thus leaves the quotient as $ 5,10,\text{ and 1} $ .
Now, again we find the common factor of 5, 10, and 1, we get
 $ \begin{align}
  & \Rightarrow 5\left| \!{\underline {\,
  5,10,1 \,}} \right. \\
 & \Rightarrow \text{ }\left| \!{\underline {\,
  1,2,1 \,}} \right. \\
\end{align} $ ----------- (2)
From above we get 5 is the common factor of 5, 10, and 1; therefore, it leaves the quotient as 1, 2, and 1.
Now, we find the common factor of 1, 2, and 1, we get
 $ \begin{align}
  & \Rightarrow 2\left| \!{\underline {\,
  1,2,1 \,}} \right. \\
 & \Rightarrow \text{ }\left| \!{\underline {\,
  1,1,1 \,}} \right. \\
\end{align} $ -------------- (3)
So, from above we see that the common factor of 1, 2, and 1 is 2 thus we get the quotient as 1, 1, and 1. So, now we stop here.
After getting the factors 2, 5, and 2 from equation (1), (2), and (3), we multiply all the factors, that is
 $ \begin{align}
  & LCM(10,20,2)=2.(5).(2) \\
 & \Rightarrow LCM(10,20,2)=20 \\
\end{align} $
Therefore, we get the least common multiple of 10, 20, and 2 is 20, which is our required answer.

Note:
 One of the alternative methods for solving this problem, is instead of using $ \left| \!{\underline {\,
  {} \,}} \right. $ , we can find the factors of individual numbers and then take common factors among them, to get the required answer.
An alternative method:
We have to find the value of $ LCM(10,20,2) $ .
No, we take factors of individual numbers, we get
Factors of 10 = $ 1X2X5={{2}^{1}}{{.5}^{1}} $ ---- (4)
Factors of 20 = $ 1X2X2X5={{2}^{2}}{{.5}^{1}} $ ---- (5)
Factors of 2 = $ 1X2={{2}^{1}} $ ---- (6)
Now, from equation (4), (5), and (6), we take the common factors, we get
2 occurs 2 times, and
5 occurs 1 time
Therefore, we get
 $ \begin{align}
  & LCM(10,20,2)={{2}^{2}}.(5) \\
 & LCM(10,20,2)=4.(5) \\
\end{align} $
 $ LCM(10,20,2)=20 $ which is our required answer.