For 378, 180 and 420, Is H.C.F × L.C.M of three numbers equal to the product of the three numbers?

First, we will find out the prime factors of 378, 180 and 420

Prime factors of 378 = 2 × 3 × 3 × 3 × 7

Prime factors of 180 = 2 × 2 × 3 × 3 × 5

Prime factors of 420 = 2 × 2 × 3 × 5 × 7

So, H.C.F of 378, 180 and 420 is 2 × 3 = 6 (being product of the least power of each common factor)

And L.C.M of 378, 180 and 420 is 2 × 2 × 3 × 3 × 3 × 5 × 7 = 3780 (being product of the highest power of the prime factors)

To check whether L.C.M (378, 180 and 420) × H.C.F (378, 180 and 420) is equal to the product of the three numbers

3780 × 6

=22680

And 378 × 180 × 420

= 28576800

⇒ 3780 × 6 ≠ 2857680

Hence L.C.M × H.C.F is not equal to the product of the three numbers.