2.1 Write the numbers 24,72 and 144 as the product of their prime factors

To express the numbers 24, 72, and 144 as the product of their prime factors, we will factor each number step by step. ### Step 1: Factor 24 1. Start with 24. 2. Divide by the smallest prime number, which is 2: - \( 24 \div 2 = 12 \) 3. Divide 12 by 2: - \( 12 \div 2 = 6 \) 4. Divide 6 by 2: - \( 6 \div 2 = 3 \) 5. Now, 3 is a prime number. So, the prime factorization of 24 is: \[ 24 = 2^3 \times 3^1 \] ### Step 2: Factor 72 1. Start with 72. 2. Divide by 2: - \( 72 \div 2 = 36 \) 3. Divide 36 by 2: - \( 36 \div 2 = 18 \) 4. Divide 18 by 2: - \( 18 \div 2 = 9 \) 5. Now, divide 9 by 3: - \( 9 \div 3 = 3 \) 6. Finally, 3 is a prime number. So, the prime factorization of 72 is: \[ 72 = 2^3 \times 3^2 \] ### Step 3: Factor 144 1. Start with 144. 2. Divide by 2: - \( 144 \div 2 = 72 \) 3. Divide 72 by 2: - \( 72 \div 2 = 36 \) 4. Divide 36 by 2: - \( 36 \div 2 = 18 \) 5. Divide 18 by 2: - \( 18 \div 2 = 9 \) 6. Now, divide 9 by 3: - \( 9 \div 3 = 3 \) 7. Finally, 3 is a prime number. So, the prime factorization of 144 is: \[ 144 = 2^4 \times 3^2 \] ### Summary of Prime Factorizations - \( 24 = 2^3 \times 3^1 \) - \( 72 = 2^3 \times 3^2 \) - \( 144 = 2^4 \times 3^2 \)