4.3.2 Determine the lowest common multiple (LCM) of 252 and 360 . \( \qquad \) You are given the following information.

To determine the **Lowest Common Multiple (LCM)** of **252** and **360**, we'll use the **prime factorization method**. Here's a step-by-step guide: ### Step 1: Prime Factorization **1. Factorize 252:** \[ 252 = 2^2 \times 3^2 \times 7 \] - \(252 \div 2 = 126\) - \(126 \div 2 = 63\) - \(63 \div 3 = 21\) - \(21 \div 3 = 7\) **2. Factorize 360:** \[ 360 = 2^3 \times 3^2 \times 5 \] - \(360 \div 2 = 180\) - \(180 \div 2 = 90\) - \(90 \div 2 = 45\) - \(45 \div 3 = 15\) - \(15 \div 3 = 5\) ### Step 2: Identify the Highest Powers of All Prime Factors List all the prime factors from both numbers and choose the highest exponent for each: - **2:** Highest power is \(2^3\) (from 360) - **3:** Highest power is \(3^2\) (common in both) - **5:** Present only in 360 as \(5^1\) - **7:** Present only in 252 as \(7^1\) ### Step 3: Multiply These Highest Powers Together \[ \text{LCM} = 2^3 \times 3^2 \times 5 \times 7 \] \[ \text{LCM} = 8 \times 9 \times 5 \times 7 \] \[ \text{LCM} = 72 \times 35 \] \[ \text{LCM} = 2520 \] ### Conclusion The **Lowest Common Multiple (LCM)** of **252** and **360** is **2520**.