Question

The product of two numbers is 2160 and their HCF is 12. Find the LCM of these numbers.

Answer 1

The product of HCF and LCM of two numbers is equal to the product of two numbers. Then substitute the values to get the value of their LCM.

Complete step-by-step answer:


We have been given the product of 2 numbers as 2160.
The HCF (highest common factor) of both these numbers is 12.
We know that product of two numbers = HCF \[\times \] LCM
where, LCM is the least number which is exactly divisible by each of the given numbers. The HCF is the highest common factor or the largest number that divides 2 or more numbers.
The product of HCF and LCM of any 2 numbers is always equal to the product of 2 numbers.
\[\therefore \] Product of 2 numbers = HCF \[\times \] LCM
\[\begin{align}
  & 2160=12\times LCM \\
 & \therefore LCM={}^{2160}/{}_{12}=180 \\
\end{align}\]
Thus we got the LCM of two numbers as 180.

Note:
If we were asked to get the 2 numbers, we can find it from the product, LCM and HCF. Consider the two numbers as 12x and12y, where x and y are prime numbers.
Thus product of 2 numbers = 2160
\[\begin{align}
  & \therefore 12x\times 12y=2160 \\
 & \therefore xy=\dfrac{2160}{12\times 12}=15 \\
\end{align}\]
Thus to get \[xy=15\], 2 possible solutions will be \[\left( 3\times 5 \right)\] and \[\left( 1\times 15 \right)\].
Thus 12x and 12y will become,
with \[\left( 3,5 \right)\Rightarrow \left( 12\times 3 \right),\left( 12\times 5 \right)=36,60\]
with \[\left( 1,15 \right)\Rightarrow \left( 12\times 1 \right),\left( 12\times 15 \right)=12,180\]
Thus the product of three numbers will be 2160.
Thus 2 pairs are \[\left( 36,60 \right)\] and \[\left( 12,180 \right)\].