Question

What is the LCM of 6 and 7?

Answer 1

Hint: To calculate the LCM of two numbers, we should first find the prime factorisation of both numbers, and then, we can write LCM by multiplying the prime factors as many times as the maximum number of times of their occurrence in any given number.

Complete step by step solution:
Least Common Multiple or LCM of two numbers, a and b, is defined as the smallest number possible that is perfectly divisible by both a and b.
To use prime factorisation method, we first calculate the prime factorisation, and then list the prime factors as many times as the maximum number of times of their occurrence.
Here in this question, we need to calculate the LCM of 6 and 7.
Prime factorisation of 6:
\[\begin{align}
  & 2\left| \!{\underline {\,
  6 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3 \,}} \right. \\
 & \text{ }\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}\]
$\therefore 6=2\times 3$
Prime factorisation of 7:
$\begin{align}
  & 7\left| \!{\underline {\,
  7 \,}} \right. \\
 & \text{ }\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}$
$\therefore 7=7\times 1$
Now, for the prime factor 2, we have,
The number of times of occurrence of 2 in prime factorisation of 6 = 1
The number of times of occurrence of 2 in prime factorisation of 7 = 0
So, the maximum number of times of occurrence of 2 = 1 …(i)
Similarly, for the prime factor 3, we have,
The number of times of occurrence of 3 in prime factorisation of 6 = 1
The number of times of occurrence of 3 in prime factorisation of 7 = 0
So, the maximum number of times of occurrence of 3 = 1 …(ii)
Similarly, for the prime factor 7, we have,
The number of times of occurrence of 7 in prime factorisation of 6 = 0
The number of times of occurrence of 7 in prime factorisation of 7 = 1
So, the maximum number of times of occurrence of 7 = 1 …(iii)
Now using (i), (ii) and (iii), we can say that 2 must occur once, 3 must also occur once, and 7 must also occur once. Thus,
$LCM=2\times 3\times 7$
$\Rightarrow LCM=42$
Hence, the LCM of 6 and 7 is 42.

Note: Here, we can notice that the two numbers 6 and 7 do not have a common factor other than 1. In such a case, the LCM will always be equal to the product of the two numbers, i.e., $LCM=6\times 7=42$ .