Find the LCM and HCF of $378,180$ and $420$ by prime factorization method. Is LCM $ \times $ HCF of $3$ numbers equal to the product of the $3$ numbers?

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Hint: Here we will find the LCM of number and we will find the HCF of the same number. After finding the LCM and HCF of that number then using the prime factorization method we will find out the answer of the given condition. Finally we get the answer to this question.

Complete step-by-step answer:
First we will find the Highest Common Factor for given numbers.
  378 = 2 \times 3 \times 3 \times 3 \\
  180 = 2 \times 2 \times 3 \times 3 \times 5 \\
  420 = 2 \times 2 \times 3 \times 5 \times 7 \\
So, finally HCF $(378,180,420) = 2 \times 3 = 6$
HCF of given number is $6$
Next we will find out the Least Common Factor of given numbers
LCM $(378,180,420) = 3780$
So, here LCM of given number is $3780$
Here we will find given condition is satisfied or not
$ \Rightarrow $ LCM $(378,180,420) \times $ HCF $(378,180,420)$ $ \ne $ $(378,180,420)$
$ \Rightarrow 3780 \times 6 \ne 378,180,420$
$\therefore 22680 \ne 28576800$
The condition is not satisfied.
That means HCF and LCM of the three numbers are not equal to the product of three numbers.

Additional information:
The H.C.F. defines the greatest factor present in between given two or more numbers, whereas L.C.M. defines the least number which is exactly divisible by two or more numbers. H.C.F. is also called the greatest common factor (GCF) and LCM is also called the Least Common Divisor. To find the HCF of two or more numbers, express each number as a product of prime numbers. The product of the least powers of common prime terms gives us the HCF.

"Highest common factor ( HCF) of two numbers is the largest number that divides evenly into both numbers. In other words the HCF is the largest of all common factors." "Lowest common multiple (LCM)is the smallest number that is a common multiple of two or more numbers."