Find the LCM between 28,36 and 12

To find the **Least Common Multiple (LCM)** of the numbers **28**, **36**, and **12**, you can use the method of prime factorization. Here's a step-by-step guide to help you understand the process: ### Step 1: Prime Factorization Break down each number into its prime factors. 1. **28** - \(28 = 2 \times 14\) - \(14 = 2 \times 7\) - **Prime Factors:** \(2^2 \times 7\) 2. **36** - \(36 = 6 \times 6\) - \(6 = 2 \times 3\) - \(6 = 2 \times 3\) - **Prime Factors:** \(2^2 \times 3^2\) 3. **12** - \(12 = 4 \times 3\) - \(4 = 2 \times 2\) - **Prime Factors:** \(2^2 \times 3\) ### Step 2: Identify the Highest Powers of All Prime Factors For each distinct prime number in the factorizations, take the highest power that appears. - **Prime Number 2:** - Highest power among the numbers: \(2^2\) - **Prime Number 3:** - Highest power among the numbers: \(3^2\) - **Prime Number 7:** - Highest power among the numbers: \(7^1\) ### Step 3: Multiply These Highest Powers Together to Find the LCM \[ \text{LCM} = 2^2 \times 3^2 \times 7 = 4 \times 9 \times 7 = 36 \times 7 = 252 \] ### Conclusion The **Least Common Multiple (LCM)** of **28**, **36**, and **12** is **252**. ### Verification To ensure that 252 is indeed the LCM, let's verify by dividing it by each of the original numbers: - \(252 \div 28 = 9\) (whole number) - \(252 \div 36 = 7\) (whole number) - \(252 \div 12 = 21\) (whole number) Since 252 is divisible by 28, 36, and 12 without any remainder, it confirms that 252 is the correct LCM.