Question

What is the LCM of two consecutive numbers?

Answer 1

Hint: We first form two consecutive numbers. We find the HCF of those numbers. We use the condition where multiplication of the HCF and the LCM of two numbers is equal to the multiplication of those numbers. This gives the solution for LCM of two consecutive numbers.

Complete step by step answer:
We first find the HCF of two consecutive numbers.
We know that the difference of two consecutive numbers is 1.
Let’s assume the numbers to be $n$ and $n+1$.
We use the successive division method to find HCF, we first need to arrange the given numbers in ascending order.
$n\overset{1}{\overline{\left){\begin{align}
  & n+1 \\
 & \underline{n} \\
 & 1 \\
\end{align}}\right.}}$
$1\overset{n}{\overline{\left){\begin{align}
  & n \\
 & \underline{n} \\
 & 0 \\
\end{align}}\right.}}$
Therefore, the HCF is 1.
Now we know that the multiplication of the HCF and the LCM of two numbers is equal to the multiplication of those numbers.
So, multiplication of $n$ and $n+1$ is $n\left( n+1 \right)$. The HCF is 1.
So, the LCM of those numbers will be $\dfrac{n\left( n+1 \right)}{1}=n\left( n+1 \right)$.
This gives that the LCM of two consecutive numbers is the multiplication of those numbers.

Note: We can also treat two consecutive numbers as co-prime numbers and we know that LCM of co-primes numbers is the multiplication of those numbers.
We take two consecutive numbers 20 and 21 for example. The HCF of 20 and 21 is 1.
The LCM of those numbers is $20\times 21=420$.