Question

# 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.

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.