Question

If LCM $ \left( {p,q,r} \right) = 420 $ , HCF $ \left( {p,q,r} \right) = 1 $ , HCF $ \left( {p,q} \right) = 3 $ , HCF $ \left( {q,r} \right) = 5 $ and $ HCF\left( {p,r} \right) = 1 $ , then the product of $ p,q{\text{ and r}} $ is
 $ \left( a \right){\text{ 630}} $
 $ \left( b \right){\text{ 6300}} $
 $ \left( c \right){\text{ 450}} $
 $ \left( d \right){\text{ 6500}} $

Answer 1

Hint:
This problem becomes very simple if we know the formula. So the formula is the main key for this problem. $ p.q.r = \left[ {\dfrac{{LCM\left( {p,q,r} \right)HCF\left( {p,q} \right)HCF\left( {p,r} \right)}}{{HCF\left( {p,q,r} \right)}}} \right] $ So by using this formula we can calculate the $ pqr $ just by taking it on one side and keeping the rest at the other side of it.

The formula used:
 $ p.q.r $ can be calculated by using the below formula
 $ p.q.r = \left[ {\dfrac{{LCM\left( {p,q,r} \right)HCF\left( {p,q} \right)HCF\left( {q,r} \right)HCF\left( {p,r} \right)}}{{HCF\left( {p,q,r} \right)}}} \right] $

Complete step by step solution:
 First of all we will see the values given to us. So the values are given as
If LCM $ \left( {p,q,r} \right) = 420 $ , HCF $ \left( {p,q,r} \right) = 1 $ , HCF $ \left( {p,q} \right) = 3 $ , HCF $ \left( {q,r} \right) = 5 $ and $ HCF\left( {p,r} \right) = 1 $
So by using the formula, we have
 $ p.q.r = \left[ {\dfrac{{LCM\left( {p,q,r} \right)HCF\left( {p,q} \right)HCF\left( {q,r} \right)HCF\left( {p,r} \right)}}{{HCF\left( {p,q,r} \right)}}} \right] $
Now on substituting the values, we get
 $ \Rightarrow \dfrac{{420 \times 3 \times 5 \times 1}}{1} $
On solving the above, we get
 $ \Rightarrow 6300 $
Therefore, $ 6300 $ will be the product of $ p,q{\text{ and r}} $ .

And hence, the option $ \left( b \right) $ is correct.

Additional information:
As we realize that the result of the most elevated regular factor (H.C.F.) and least normal numerous (L.C.M.) of two numbers is equivalent to the result of two numbers.

The HCF of at least two numbers is the biggest regular factor of the given numbers. It is the best number that isolates at least two given numbers. While The LCM of at least two numbers is the most modest number that is a typical difference of at least two numbers. It is the most modest number which is distinguishable by at least two given numbers.

Note:
This type of question can only be solved by using the formula and if we know it then it can be solved very easily. HCF and LCM have a vast use in our real life too. For example to split the things into some smaller sections HCF can be used similarly to know about the event that is or will be repeating continually, for this LCM can be used.