1. Let's start by understanding the problem: We want to find the Highest Common Factor (HCF) and Least Common Multiple (LCM) of given numbers. 2. Case 1: Find HCF and LCM of two numbers, for example, 12 and 18. - Prime factorization of 12: $$2^2 \times 3$$ - Prime factorization of 18: $$2 \times 3^2$$ - HCF is the product of the lowest powers of common primes: $$2^1 \times 3^1 = 6$$ - LCM is the product of the highest powers of all primes: $$2^2 \times 3^2 = 36$$ 3. Case 2: Find HCF and LCM of three numbers, for example, 8, 12, and 20. - Prime factorization of 8: $$2^3$$ - Prime factorization of 12: $$2^2 \times 3$$ - Prime factorization of 20: $$2^2 \times 5$$ - HCF is the product of the lowest powers of common primes: $$2^2 = 4$$ - LCM is the product of the highest powers of all primes: $$2^3 \times 3 \times 5 = 120$$ 4. Case 3: When one number divides the other, for example, 15 and 45. - Since 15 divides 45, HCF is 15. - LCM is the larger number, 45. 5. Summary: To find HCF, take the product of the smallest powers of common prime factors. To find LCM, take the product of the largest powers of all prime factors involved. This method works for any set of integers.