Find the LCM and HCF of the following pairs of integers and verify that LCM $\times$ HCF $=$ Product of the integers:
(i) 26 and 91

Given:

Given pair of integers is 26 and 91.

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 26:

• $2\ \times\ 13\ =\ 2^1\ \times\ 13^1$

Prime factorisation of 91:

• $7\ \times\ 13\ =\ 7^1\ \times\ 13^1$

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

$2^1\ \times\ 13^1\ \times\ 7^1\ =\ 182$

LCM(26, 91)  $=$  182

Multiplying all common prime factors: 

$13^1\ =\ 13$

HCF(26, 91)  $=$  13

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

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

182 $\times$ 13 $=$ 26 $\times$ 91

2366 $=$ 2366.