Find the LCM and HCF of the following pairs of integers and verify that LCM $\times$ HCF $=$ product of two numbers:
(i) 26 and 91
(ii) 510 and 92
(iii) 336 and 54.

To do:

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

Solution: 

Calculating LCM and HCF using prime factorization method:

Writing the numbers as a product of their prime factors:

(i) 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.

(ii) 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. 

(iii) 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.