e. 456 and 572 1. 205 and 315 \( \qquad \) \( \square \) 3. Calculate the LCM using factorisation or inspection. \[ \begin{array}{l} \text { Factors of } 123 \text { and } 141 \\ \begin{array}{r|r|ll} 123 & 3 & 141 & 3 \end{array} l \\ 41 \\ 41 \end{array} \] Example: Factors of 123 and 141 a. 243 and 729 b. 200 and 1000

Sure, let's calculate the **Least Common Multiple (LCM)** for the given pairs of numbers using **prime factorization**. The LCM of two numbers is the smallest number that is a multiple of both numbers. We'll go through each pair step-by-step. --- ## Example: Factors of 123 and 141 Before we tackle the main problems, let's understand the example provided. ### Step 1: Prime Factorization 1. **Factorize 123** - 123 ÷ 3 = 41 - 41 is a prime number. - **Prime Factors of 123:** \(3 \times 41\) 2. **Factorize 141** - 141 ÷ 3 = 47 - 47 is a prime number. - **Prime Factors of 141:** \(3 \times 47\) ### Step 2: Determine the Highest Powers of All Prime Factors - **Prime Factors Involved:** 3, 41, 47 - **Highest Powers:** - \(3^1\) - \(41^1\) - \(47^1\) ### Step 3: Calculate the LCM \[ \text{LCM} = 3 \times 41 \times 47 = 5,781 \] --- Now, let's apply the same method to the given problems. --- ## Problem e. LCM of 456 and 572 ### Step 1: Prime Factorization 1. **Factorize 456** - 456 ÷ 2 = 228 - 228 ÷ 2 = 114 - 114 ÷ 2 = 57 - 57 ÷ 3 = 19 - 19 is a prime number. - **Prime Factors of 456:** \(2^3 \times 3 \times 19\) 2. **Factorize 572** - 572 ÷ 2 = 286 - 286 ÷ 2 = 143 - 143 ÷ 11 = 13 - 13 is a prime number. - **Prime Factors of 572:** \(2^2 \times 11 \times 13\) ### Step 2: Determine the Highest Powers of All Prime Factors - **Prime Factors Involved:** 2, 3, 11, 13, 19 - **Highest Powers:** - \(2^3\) (from 456) - \(3^1\) (from 456) - \(11^1\) (from 572) - \(13^1\) (from 572) - \(19^1\) (from 456) ### Step 3: Calculate the LCM \[ \text{LCM} = 2^3 \times 3 \times 11 \times 13 \times 19 = 8 \times 3 \times 11 \times 13 \times 19 \] \[ \text{LCM} = 24 \times 11 = 264 \] \[ 264 \times 13 = 3,432 \] \[ 3,432 \times 19 = 65,208 \] \[ \boxed{65,\!208} \] --- ## Problem 1. LCM of 205 and 315 ### Step 1: Prime Factorization 1. **Factorize 205** - 205 ÷ 5 = 41 - 41 is a prime number. - **Prime Factors of 205:** \(5 \times 41\) 2. **Factorize 315** - 315 ÷ 5 = 63 - 63 ÷ 3 = 21 - 21 ÷ 3 = 7 - 7 is a prime number. - **Prime Factors of 315:** \(3^2 \times 5 \times 7\) ### Step 2: Determine the Highest Powers of All Prime Factors - **Prime Factors Involved:** 3, 5, 7, 41 - **Highest Powers:** - \(3^2\) (from 315) - \(5^1\) (common to both) - \(7^1\) (from 315) - \(41^1\) (from 205) ### Step 3: Calculate the LCM \[ \text{LCM} = 3^2 \times 5 \times 7 \times 41 = 9 \times 5 \times 7 \times 41 \] \[ \text{LCM} = 45 \times 7 = 315 \] \[ 315 \times 41 = 12,915 \] \[ \boxed{12,\!915} \] --- ## Example a. LCM of 243 and 729 ### Step 1: Prime Factorization 1. **Factorize 243** - \(243 = 3^5\) 2. **Factorize 729** - \(729 = 3^6\) ### Step 2: Determine the Highest Powers of All Prime Factors - **Prime Factors Involved:** 3 - **Highest Power:** - \(3^6\) (from 729) ### Step 3: Calculate the LCM \[ \text{LCM} = 3^6 = 729 \] \[ \boxed{729} \] --- ## Example b. LCM of 200 and 1000 ### Step 1: Prime Factorization 1. **Factorize 200** - 200 ÷ 2 = 100 - 100 ÷ 2 = 50 - 50 ÷ 2 = 25 - 25 ÷ 5 = 5 - 5 ÷ 5 = 1 - **Prime Factors of 200:** \(2^3 \times 5^2\) 2. **Factorize 1000** - 1000 ÷ 2 = 500 - 500 ÷ 2 = 250 - 250 ÷ 2 = 125 - 125 ÷ 5 = 25 - 25 ÷ 5 = 5 - 5 ÷ 5 = 1 - **Prime Factors of 1000:** \(2^3 \times 5^3\) ### Step 2: Determine the Highest Powers of All Prime Factors - **Prime Factors Involved:** 2, 5 - **Highest Powers:** - \(2^3\) (common to both) - \(5^3\) (from 1000) ### Step 3: Calculate the LCM \[ \text{LCM} = 2^3 \times 5^3 = 8 \times 125 = 1,000 \] \[ \boxed{1,\!000} \] --- ## Summary of LCM Calculations 1. **LCM of 456 and 572:** **65,208** 2. **LCM of 205 and 315:** **12,915** 3. **LCM of 243 and 729:** **729** 4. **LCM of 200 and 1000:** **1,000** --- If you have any more pairs you'd like to calculate or need further clarification on the steps, feel free to ask!