Question 3 \( 3.1 .1) \) Prime factors of 132 3.1 .2 HCF of 300 and 120 3.1 .3 LCM of 30 and 135

To find the prime factors of 132, you can break it down by starting with the smallest prime number. Dividing 132 by 2 (the smallest prime) gives you 66, then divide 66 by 2 again to get 33. Next, 33 can be divided by 3 to get 11. Since 11 is a prime number, the prime factorization of 132 is \( 2^2 \times 3 \times 11 \). For the HCF (Highest Common Factor) of 300 and 120, first, determine their prime factorizations. 300 breaks down to \( 2^2 \times 3 \times 5^2 \) and 120 to \( 2^3 \times 3 \times 5 \). The HCF is found by taking the lowest power of common prime factors. Thus, the HCF of 300 and 120 is \( 2^2 \times 3 \times 5 = 60 \). The LCM (Lowest Common Multiple) of 30 and 135 can also be determined through their prime factorizations. 30 breaks down to \( 2 \times 3 \times 5 \) and 135 to \( 3^3 \times 5 \). Take the highest powers of each prime: \( 2^1 \), \( 3^3 \), and \( 5^1 \). With these, the LCM of 30 and 135 is \( 2^1 \times 3^3 \times 5^1 = 270 \).