Use prime factorisation to find the HCF and LCM of 24 and 32. Hence, verify that HCF × LCM = product of two numbers.

Given,

Numbers = 24 and 32

Prime factorisation of 24 and 32:

$\begin{array}{l|r} 2 & 24 \ \hline 2 & 12 \ \hline 2 & 6 \ \hline 3 & 3 \ \hline & 1 \end{array} \quad \text{ and } \quad \begin{array}{l|r} 2 & 32 \ \hline 2 & 16 \ \hline 2 & 8 \ \hline 2 & 4 \ \hline 2 & 2 \ \hline & 1 \end{array}$

⇒ 24 = 2 × 2 × 2 × 3 = 2³ × 3

⇒ 32 = 2 × 2 × 2 × 2 × 2 = 2⁵

HCF = product of the common prime factors with the lowest powers = 2³ = 8

LCM = product of all prime factors with the highest powers = 2⁵ × 3 = 32 × 3 = 96

Verification:

HCF × LCM = 8 × 96 = 768

Product of the two numbers = 24 × 32 = 768

Since 768 = 768, HCF × LCM = product of the two numbers is verified.