2 Determine the LCM of 882 and 840

To determine the **Least Common Multiple (LCM)** of **882** and **840**, follow these steps: 1. **Prime Factorization** 2. **Identify the Highest Powers of Each Prime** 3. **Multiply These Prime Powers Together** Let's break it down: --- ### 1. Prime Factorization **Factorizing 882:** 1. **Divide by 2:** 882 ÷ 2 = 441 So, 882 = 2 × 441 2. **Factorize 441:** 441 ÷ 3 = 147 147 ÷ 3 = 49 49 = 7 × 7 So, 441 = 3² × 7² Therefore, 882 = 2 × 3² × 7² **Factorizing 840:** 1. **Divide by 2 repeatedly:** 840 ÷ 2 = 420 420 ÷ 2 = 210 210 ÷ 2 = 105 So, 840 = 2³ × 105 2. **Factorize 105:** 105 ÷ 3 = 35 35 ÷ 5 = 7 So, 105 = 3 × 5 × 7 Therefore, 840 = 2³ × 3 × 5 × 7 --- ### 2. Identify the Highest Powers of Each Prime List all the prime numbers found in the factorizations: **2, 3, 5, 7** - **2:** - 882 has 2¹ - 840 has 2³ - **Highest Power:** 2³ - **3:** - 882 has 3² - 840 has 3¹ - **Highest Power:** 3² - **5:** - 882 has 5⁰ (since 5 is not a factor) - 840 has 5¹ - **Highest Power:** 5¹ - **7:** - 882 has 7² - 840 has 7¹ - **Highest Power:** 7² --- ### 3. Multiply These Prime Powers Together \[ \text{LCM} = 2³ \times 3² \times 5¹ \times 7² \] Calculate step-by-step: 1. \(2³ = 8\) 2. \(3² = 9\) 3. \(5¹ = 5\) 4. \(7² = 49\) Now, multiply them together: \[ 8 \times 9 = 72 \\ 72 \times 5 = 360 \\ 360 \times 49 = 17,640 \] --- ### **Final Answer** The **LCM of 882 and 840 is 17,640**.