\( =1 \) P) EXERCISE 2 \( \begin{array}{ll}\text { (a) Consider the numbers } 24 \text { and } 504 . & \text { (b) Consider the numbers } 72 \text { and } 180 \text {. } \\ \text { (1) Write both numbers as a product of primes. } & \text { (1) Write both numbers as a product of primes. } \\ \text { (2) Is } 24 \text { a factor of } 504 \text { ? } & \text { (2) Is } 180 \text { a multiple of } 72 \text { ? }\end{array} \)

Certainly! Let's tackle **EXERCISE 2** step by step. --- ### **(a) Consider the numbers 24 and 504.** #### **(1) Write both numbers as a product of primes.** To express each number as a product of prime factors, we'll perform prime factorization. **For 24:** 1. Divide by the smallest prime number (2): \[ 24 \div 2 = 12 \] 2. Continue dividing by 2: \[ 12 \div 2 = 6 \] 3. Divide by 2 again: \[ 6 \div 2 = 3 \] 4. Finally, divide by 3: \[ 3 \div 3 = 1 \] So, the prime factorization of **24** is: \[ 24 = 2^3 \times 3 \] **For 504:** 1. Divide by 2: \[ 504 \div 2 = 252 \] 2. Divide by 2 again: \[ 252 \div 2 = 126 \] 3. Divide by 2 once more: \[ 126 \div 2 = 63 \] 4. Now, divide by 3: \[ 63 \div 3 = 21 \] 5. Divide by 3 again: \[ 21 \div 3 = 7 \] 6. Finally, divide by 7: \[ 7 \div 7 = 1 \] So, the prime factorization of **504** is: \[ 504 = 2^3 \times 3^2 \times 7 \] #### **(2) Is 24 a factor of 504?** To determine if **24** is a factor of **504**, we can check if the prime factors of 24 are all present in the prime factorization of 504 with at least the same exponents. - **Prime factors of 24:** \(2^3 \times 3\) - **Prime factors of 504:** \(2^3 \times 3^2 \times 7\) Comparing the exponents: - For prime number 2: \(2^3\) in both. - For prime number 3: \(3^1\) in 24 vs. \(3^2\) in 504. Since 504 has all the prime factors of 24 with equal or higher exponents, **24 is indeed a factor of 504**. --- ### **(b) Consider the numbers 72 and 180.** #### **(1) Write both numbers as a product of primes.** **For 72:** 1. Divide by 2: \[ 72 \div 2 = 36 \] 2. Divide by 2 again: \[ 36 \div 2 = 18 \] 3. Divide by 2 once more: \[ 18 \div 2 = 9 \] 4. Now, divide by 3: \[ 9 \div 3 = 3 \] 5. Finally, divide by 3: \[ 3 \div 3 = 1 \] So, the prime factorization of **72** is: \[ 72 = 2^3 \times 3^2 \] **For 180:** 1. Divide by 2: \[ 180 \div 2 = 90 \] 2. Divide by 2 again: \[ 90 \div 2 = 45 \] 3. Now, divide by 3: \[ 45 \div 3 = 15 \] 4. Divide by 3 again: \[ 15 \div 3 = 5 \] 5. Finally, divide by 5: \[ 5 \div 5 = 1 \] So, the prime factorization of **180** is: \[ 180 = 2^2 \times 3^2 \times 5 \] #### **(2) Is 180 a multiple of 72?** To determine if **180** is a multiple of **72**, we check if 72 divides 180 without leaving a remainder. Alternatively, we can compare their prime factorizations: - **Prime factors of 72:** \(2^3 \times 3^2\) - **Prime factors of 180:** \(2^2 \times 3^2 \times 5\) For 180 to be a multiple of 72, it must have at least the same number of each prime factor as 72. - For prime number 2: \(2^3\) in 72 vs. \(2^2\) in 180. - For prime number 3: \(3^2\) in both. - Additional prime number 5 in 180. Since 180 has fewer factors of 2 than 72, **180 is not a multiple of 72**. --- ### **Summary of Answers** **(a)** 1. \(24 = 2^3 \times 3\) \(504 = 2^3 \times 3^2 \times 7\) 2. Yes, **24 is a factor of 504**. **(b)** 1. \(72 = 2^3 \times 3^2\) \(180 = 2^2 \times 3^2 \times 5\) 2. No, **180 is not a multiple of 72**.