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

**Step 3.1.1: Prime factors of 132** First, divide 132 by \(2\): \[ 132 \div 2 = 66 \] Divide 66 by \(2\) again: \[ 66 \div 2 = 33 \] Now, divide 33 by \(3\): \[ 33 \div 3 = 11 \] Since 11 is a prime number, the prime factorization of 132 is: \[ 132 = 2^2 \times 3 \times 11 \] --- **Step 3.1.2: HCF of 300 and 120** First, express each number as a product of prime factors. For 300: \[ 300 = 2^2 \times 3 \times 5^2 \] For 120: \[ 120 = 2^3 \times 3 \times 5 \] The highest common factor (HCF) is determined by taking the lowest power of the common primes: - For \(2\): lowest power is \(2^2\) - For \(3\): lowest power is \(3^1\) - For \(5\): lowest power is \(5^1\) Thus, the HCF is: \[ \text{HCF} = 2^2 \times 3 \times 5 = 4 \times 3 \times 5 = 60 \] --- **Step 3.1.3: LCM of 30 and 135** Express each number as a product of prime factors. For 30: \[ 30 = 2 \times 3 \times 5 \] For 135: \[ 135 = 3^3 \times 5 \] The least common multiple (LCM) is determined by taking the highest power of all the primes present: - For \(2\): highest power is \(2^1\) (appears only in 30) - For \(3\): highest power is \(3^3\) - For \(5\): highest power is \(5^1\) Thus, the LCM is: \[ \text{LCM} = 2 \times 3^3 \times 5 = 2 \times 27 \times 5 = 270 \]