Find the LCM of 48, 72, 140.

Answer

To find the Least Common Multiple (LCM) of the numbers 48, 72, and 140, we will follow these steps:

### Step 1: Prime Factorization
First, we need to find the prime factorization of each number.

**For 48:**
- 48 can be divided by 2:
- \(48 \div 2 = 24\)
- 24 can be divided by 2:
- \(24 \div 2 = 12\)
- 12 can be divided by 2:
- \(12 \div 2 = 6\)
- 6 can be divided by 2:
- \(6 \div 2 = 3\)
- 3 is a prime number.

So, the prime factorization of 48 is:
\[ 48 = 2^4 \times 3^1 \]

**For 72:**
- 72 can be divided by 2:
- \(72 \div 2 = 36\)
- 36 can be divided by 2:
- \(36 \div 2 = 18\)
- 18 can be divided by 2:
- \(18 \div 2 = 9\)
- 9 can be divided by 3:
- \(9 \div 3 = 3\)
- 3 is a prime number.

So, the prime factorization of 72 is:
\[ 72 = 2^3 \times 3^2 \]

**For 140:**
- 140 can be divided by 2:
- \(140 \div 2 = 70\)
- 70 can be divided by 2:
- \(70 \div 2 = 35\)
- 35 can be divided by 5:
- \(35 \div 5 = 7\)
- 7 is a prime number.

So, the prime factorization of 140 is:
\[ 140 = 2^2 \times 5^1 \times 7^1 \]

### Step 2: Identify the Highest Powers
Next, we will identify the highest powers of each prime factor from the factorizations:

- For \(2\): The highest power is \(2^4\) (from 48).
- For \(3\): The highest power is \(3^2\) (from 72).
- For \(5\): The highest power is \(5^1\) (from 140).
- For \(7\): The highest power is \(7^1\) (from 140).

### Step 3: Calculate the LCM
Now, we will multiply these highest powers together to find the LCM:
\[ LCM = 2^4 \times 3^2 \times 5^1 \times 7^1 \]

Calculating this step by step:
1. \(2^4 = 16\)
2. \(3^2 = 9\)
3. \(5^1 = 5\)
4. \(7^1 = 7\)

Now, multiply these values together:
\[ LCM = 16 \times 9 \times 5 \times 7 \]

Calculating:
- First, \(16 \times 9 = 144\)
- Next, \(144 \times 5 = 720\)
- Finally, \(720 \times 7 = 5040\)

So, the LCM of 48, 72, and 140 is:
\[ \boxed{5040} \]