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

510 and 92

Given:

Given pair of integers is 510 and 92.

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 510:

• $2\ \times\ 3\ \times\ 5\ \times\ 17\ =\ 2^1\ \times\ 3^1\ \times\ 5^1\ \times\ 17^1$

Prime factorisation of 92:

• $2\ \times\ 2\ \times\ 23\ =\ 2^2\ \times\ 23^1$

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

$2^2\ \times\ 3^1\ \times\ 5^1\ \times\ 17^1\ \times\ 23^1\ =\ 23460$

LCM(510, 92)  $=$  23460

Multiplying all common prime factors: 

$2^1\ =\ 2$

HCF(510, 92)  $=$  2

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

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

23460 $\times$ 2 $=$ 510 $\times$ 92

46920 $=$ 46920.