Exercise 1.9 1. Find the LCM of: \( \begin{array}{ll}\text { 1.1. } 2 ; 3 & \text { 1.2. } 5 ; 7 \\ \text { 1.3. } 9 ; 81 & \text { 1.4. } 4 ; 8 ; 16 \\ \text { 1.5. } 10 ; 100 ; 1000 & \end{array} \)

Sure, let's find the **Least Common Multiple (LCM)** for each pair or set of numbers listed in Exercise 1.9. The LCM of two or more integers is the smallest positive integer that is divisible by each of the numbers. We'll use **prime factorization** as the method to determine the LCM for each set of numbers. --- ### 1.1. LCM of 2 and 3 **Prime Factorization:** - **2** is a prime number: \(2 = 2\) - **3** is a prime number: \(3 = 3\) **LCM Calculation:** - Identify all the unique prime factors. - Take the highest power of each prime factor present in the numbers. \[ \text{LCM} = 2 \times 3 = 6 \] **Answer:** 6 --- ### 1.2. LCM of 5 and 7 **Prime Factorization:** - **5** is a prime number: \(5 = 5\) - **7** is a prime number: \(7 = 7\) **LCM Calculation:** - Both numbers are distinct primes. \[ \text{LCM} = 5 \times 7 = 35 \] **Answer:** 35 --- ### 1.3. LCM of 9 and 81 **Prime Factorization:** - **9** is \(3^2\) - **81** is \(3^4\) **LCM Calculation:** - Take the highest power of 3 from both numbers. \[ \text{LCM} = 3^4 = 81 \] **Answer:** 81 --- ### 1.4. LCM of 4, 8, and 16 **Prime Factorization:** - **4** is \(2^2\) - **8** is \(2^3\) - **16** is \(2^4\) **LCM Calculation:** - Take the highest power of 2 from all numbers. \[ \text{LCM} = 2^4 = 16 \] **Answer:** 16 --- ### 1.5. LCM of 10, 100, and 1000 **Prime Factorization:** - **10** is \(2 \times 5\) - **100** is \(2^2 \times 5^2\) - **1000** is \(2^3 \times 5^3\) **LCM Calculation:** - Take the highest power of each prime factor (2 and 5). \[ \text{LCM} = 2^3 \times 5^3 = 8 \times 125 = 1000 \] **Answer:** 1000 --- ### Summary of Answers 1. **LCM of 2 and 3:** 6 2. **LCM of 5 and 7:** 35 3. **LCM of 9 and 81:** 81 4. **LCM of 4, 8, and 16:** 16 5. **LCM of 10, 100, and 1000:** 1000 Feel free to ask if you need further explanations or assistance with other problems!