Question

Show that the product of two numbers 60 and 84 is equal to the product of their HCF and LCM.

Answer 1

Hint: We start solving the problem by recalling the definitions of LCM of HCF of two or more numbers. We then find the LCM of the given numbers 60 and 84. We then factorize the numbers 60 and 84, and find the product of the common factors which is the HCF of those numbers. We then find the product of LCM and HCF, also the product of the two given numbers. We then compare both to get the required result.

Complete step by step answer:
According to the problem, we need to show that the product of two numbers 60 and 84 is equal to the product of their HCF and LCM.
Let us first recall the definitions of HCF and LCM.
Least Common Multiple (LCM) is defined as the smallest positive number that can be multiplied with the two or more given numbers.
Highest Common Factor (HCF) is defined as the greatest factor that can divide the given two or more numbers.
So, let us first find the LCM of given numbers 60 and 84.
$\begin{align}
  & 2\left| \!{\underline {\,
  60,84 \,}} \right. \\
 & 2\left| \!{\underline {\,
  30,42 \,}} \right. \\
 & 3\left| \!{\underline {\,
  15,21 \,}} \right. \\
 & \text{ }\left| \!{\underline {\,
  5,7 \,}} \right. \\
\end{align}$.
So, the LCM of the numbers 60 and 84 is $2\times 2\times 3\times 5\times 7=420$ ---(1).
Now, let us factorize the given numbers 60 and 84. We get factorization as follows.
\[\Rightarrow 60=2\times 30=2\times 2\times 15=2\times 2\times 3\times 5\].
\[\Rightarrow 84=2\times 42=2\times 2\times 21=2\times 2\times 3\times 7\].
From factorization we can see that the numbers 60 and 84 have common factors 2, 2, 3.
So, let us multiply them to get HCF. So, we get HCF = $2\times 2\times 3=12$ ---(2).
Let us multiply the HCF and LCM obtained in equations (1) and (2).
So, we get HCF $\times $ LCM = $420\times 12=5040$ ---(3).
Now, let us find the product of the numbers 60 and 84.
So, the product is $60\times 84=5040$ ---(4).

From equations (3) and (4), we can see that the product of HCF and LCM of the numbers 60 and 84 is equal to their product.

Note: We can see that the problems required good precision in calculation so, we need to perform each step carefully while solving this problem. We should not just say that one of the factors of the HCF as we need the Highest Common Factor. We can also find the ratio of the ratio of LCM and HCF from the obtained numbers. Similarly, we can expect problems to find the numbers that were divisible by both 60 and 84 that were lying between 5000 and 10000.