Find the HCF of 120 and 300 using prime factors.

To find the Highest Common Factor (HCF) of 120 and 300 using prime factors, we will follow these steps: 1. **Find the prime factorization of each number.** 2. **Identify the common prime factors.** 3. **Multiply the common prime factors to get the HCF.** ### Step 1: Prime Factorization **For 120:** - Divide by 2: \( 120 \div 2 = 60 \) - Divide by 2: \( 60 \div 2 = 30 \) - Divide by 2: \( 30 \div 2 = 15 \) - Divide by 3: \( 15 \div 3 = 5 \) - Divide by 5: \( 5 \div 5 = 1 \) So, the prime factorization of 120 is: \[ 120 = 2^3 \times 3^1 \times 5^1 \] **For 300:** - Divide by 2: \( 300 \div 2 = 150 \) - Divide by 2: \( 150 \div 2 = 75 \) - Divide by 3: \( 75 \div 3 = 25 \) - Divide by 5: \( 25 \div 5 = 5 \) - Divide by 5: \( 5 \div 5 = 1 \) So, the prime factorization of 300 is: \[ 300 = 2^2 \times 3^1 \times 5^2 \] ### Step 2: Identify Common Prime Factors Now, we will identify the common prime factors from both factorizations: - For \(2\): The minimum power is \(2^2\). - For \(3\): The minimum power is \(3^1\). - For \(5\): The minimum power is \(5^1\). ### Step 3: Multiply the Common Prime Factors Now, we multiply the common prime factors: \[ \text{HCF} = 2^2 \times 3^1 \times 5^1 \] Calculating this: \[ \text{HCF} = 4 \times 3 \times 5 \] \[ = 12 \times 5 = 60 \] Thus, the HCF of 120 and 300 is \( \boxed{60} \).