Question

What are the least common multiple of 7, 14 and 21?

Answer 1

Hint: To obtain the least common multiple of a given number we will use the common division method. Firstly we will write all the numbers inside the division sign and then we will divide them by prime numbers till we get all remainder as 1. Then we will multiply all the prime factors and get our least common multiple.

Complete step by step solution:
The numbers are given as 7, 14 and 21.
Now use common division method as below:
$\begin{align}
  & 2\left| \!{\underline {\,
  7,14,21 \,}} \right. \\
 & 3\left| \!{\underline {\,
  7,7,21 \,}} \right. \\
 & 7\left| \!{\underline {\,
  7,7,7 \,}} \right. \\
 & 1,1,1 \\
\end{align}$
The factors obtained are the prime number on the left side which is:
$\left\{ 2,3,7 \right\}$
Next we will multiply all the factors obtained as:
Least common multiple $=2\times 3\times 7$
Least common multiple $=42$
Hence least common multiple of 7, 14 and 21 is 42.

Note: Least common multiple also known as L.C.M of two or more numbers is the smallest positive integer divided by all the numbers completely. L.C.M is most commonly used when we add or subtract fraction values because we need to make the denominator same for all values for simplifying it further. Another method that can be used to find the L.C.M is by prime factoring all the values and then multiplying the factor with highest values from all the numbers together as:
Prime factor of the numbers are:
$7=1\times 7$
$14=1\times 2\times 7$
$21=1\times 3\times 7$
So taking factor 1, 2, 3 and 7 with highest power we get,
$\begin{align}
  & \Rightarrow 1\times 2\times 3\times 7 \\
 & \Rightarrow 42 \\
\end{align}$
So the least common multiple is obtained as 42 which is the same as the one we got by using the common division method.