Find the LCM and HCF of the following pairs of integers and verify that LCM $\times$ HCF $=$ Product of the integers:

336 and 54

Given:

Given pair of integers is 336 and 54.

To do:

Here we have to find the LCM and HCF of the given pair of integers and then verify that LCM $\times$ HCF $=$ Product of the integers.

Solution: 

Calculating LCM and HCF using prime factorization method:

Writing the numbers as a product of their prime factors:

Prime factorisation of 336:

• $2\ \times\ 2\ \times\ 2\ \times\ 2\ \times\ 3\ \times\ 7\ =\ 2^4\ \times\ 3^1\ \times\ 7^1$

Prime factorisation of 54:

• $2\ \times\ 3\ \times\ 3\ \times\ 3\ =\ 2^1\ \times\ 3^3$

Multiplying the highest power of each prime number these values together:

$2^4\ \times\ 3^3\ \times\ 7^1\ =\ 3024$

LCM(336, 54)  $=$  3024

Multiplying all common prime factors: 

$2^1\ \times\ 3^1\ =\ 6$

HCF(336, 54)  $=$  6

Now, verifying that LCM $\times$ HCF $=$ Product of the integers:

LCM $\times$ HCF $=$ Product of the integers

3024 $\times$ 6 $=$ 336 $\times$ 54

18144 $=$ 18144.