Find the LCM of 72, 288 and 1080.

Options

A. 4230
B. 4320
C. 2430
D. 2340

Answer

To find the LCM (Least Common Multiple) of the numbers 72, 288, and 1080, we can use the division method. Here’s a step-by-step solution:

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

1. **For 72:**
- 72 is even, divide by 2: \( 72 \div 2 = 36 \)
- 36 is even, divide by 2: \( 36 \div 2 = 18 \)
- 18 is even, divide by 2: \( 18 \div 2 = 9 \)
- 9 is divisible by 3: \( 9 \div 3 = 3 \)
- 3 is divisible by 3: \( 3 \div 3 = 1 \)

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

2. **For 288:**
- 288 is even, divide by 2: \( 288 \div 2 = 144 \)
- 144 is even, divide by 2: \( 144 \div 2 = 72 \)
- 72 is even, divide by 2: \( 72 \div 2 = 36 \)
- 36 is even, divide by 2: \( 36 \div 2 = 18 \)
- 18 is even, divide by 2: \( 18 \div 2 = 9 \)
- 9 is divisible by 3: \( 9 \div 3 = 3 \)
- 3 is divisible by 3: \( 3 \div 3 = 1 \)

So, the prime factorization of 288 is:
\[ 288 = 2^5 \times 3^2 \]

3. **For 1080:**
- 1080 is even, divide by 2: \( 1080 \div 2 = 540 \)
- 540 is even, divide by 2: \( 540 \div 2 = 270 \)
- 270 is even, divide by 2: \( 270 \div 2 = 135 \)
- 135 is divisible by 3: \( 135 \div 3 = 45 \)
- 45 is divisible by 3: \( 45 \div 3 = 15 \)
- 15 is divisible by 3: \( 15 \div 3 = 5 \)
- 5 is a prime number.

So, the prime factorization of 1080 is:
\[ 1080 = 2^3 \times 3^3 \times 5^1 \]

### Step 2: Determine the LCM
To find the LCM, we take the highest power of each prime factor from the factorizations:

- For \(2\): the highest power is \(2^5\) (from 288).
- For \(3\): the highest power is \(3^3\) (from 1080).
- For \(5\): the highest power is \(5^1\) (from 1080).

Thus, the LCM is:
\[ LCM = 2^5 \times 3^3 \times 5^1 \]

### Step 3: Calculate the LCM
Now we calculate the LCM:
1. Calculate \(2^5 = 32\)
2. Calculate \(3^3 = 27\)
3. Calculate \(5^1 = 5\)

Now, multiply these together:
\[ LCM = 32 \times 27 \times 5 \]

Calculating step-by-step:
- First, \(32 \times 27 = 864\)
- Then, \(864 \times 5 = 4320\)

Thus, the LCM of 72, 288, and 1080 is:
\[ \text{LCM} = 4320 \]

### Final Answer
The LCM of 72, 288, and 1080 is **4320**.

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