Final answer:
To find the LCM and HCF of 180 and 240 using prime factorization, we determine the highest and lowest powers of each prime factor that appear in both numbers. The LCM is the product of the highest powers, while the HCF is the product of the lowest powers.
Step-by-step explanation:
To determine the least common multiple (LCM) and highest common factor (HCF) of 180 and 240 using prime factorization, we need to find the prime factors of both numbers.
Prime factorization of 180: 2 x 2 x 3 x 3 x 5
Prime factorization of 240: 2 x 2 x 2 x 2 x 3 x 5
The LCM is found by taking the highest power of each prime factor that appears in either number. In this case, we have 2^4, 3^2, and 5^1. So, the LCM of 180 and 240 is 2^4 x 3^2 x 5 = 1440.
The HCF is found by taking the lowest power of each prime factor that appears in both numbers. In this case, we have 2^2, 3^1, and 5^0. So, the HCF of 180 and 240 is 2^2 x 3 x 1 = 12.