1. Let's start by understanding the problem: You want to learn about LCM (Least Common Multiple) and HCF (Highest Common Factor). 2. Definitions: - LCM of two numbers is the smallest number that is a multiple of both. - HCF (or GCD) of two numbers is the largest number that divides both without leaving a remainder. 3. Important formulas and rules: - For two numbers $a$ and $b$, the relationship between LCM and HCF is: $$\text{LCM}(a,b) \times \text{HCF}(a,b) = a \times b$$ - To find HCF, use prime factorization and take the product of common prime factors with the smallest powers. - To find LCM, use prime factorization and take the product of all prime factors with the highest powers. 4. Example: Find LCM and HCF of 12 and 18. - Prime factors of 12: $2^2 \times 3$ - Prime factors of 18: $2 \times 3^2$ - HCF: Take common primes with smallest powers: $2^1 \times 3^1 = 6$ - LCM: Take all primes with highest powers: $2^2 \times 3^2 = 36$ 5. Verify using formula: $$\text{LCM} \times \text{HCF} = 36 \times 6 = 216$$ $$a \times b = 12 \times 18 = 216$$ Both sides equal, so the calculations are correct. This is how you find LCM and HCF step-by-step.