The LCM of 112, 72 and 90 is

Options

A. 7560
B. 2520
C. 5040
D. 3780

Answer

To find the LCM (Least Common Multiple) of the numbers 112, 72, and 90, we will follow these steps:

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

- **112**:
- Divide by 2: \(112 \div 2 = 56\)
- Divide by 2: \(56 \div 2 = 28\)
- Divide by 2: \(28 \div 2 = 14\)
- Divide by 2: \(14 \div 2 = 7\)
- 7 is a prime number.

So, the prime factorization of 112 is \(2^4 \times 7^1\).

- **72**:
- Divide by 2: \(72 \div 2 = 36\)
- Divide by 2: \(36 \div 2 = 18\)
- Divide by 2: \(18 \div 2 = 9\)
- Divide by 3: \(9 \div 3 = 3\)
- Divide by 3: \(3 \div 3 = 1\)

So, the prime factorization of 72 is \(2^3 \times 3^2\).

- **90**:
- Divide by 2: \(90 \div 2 = 45\)
- Divide by 3: \(45 \div 3 = 15\)
- Divide by 3: \(15 \div 3 = 5\)
- 5 is a prime number.

So, the prime factorization of 90 is \(2^1 \times 3^2 \times 5^1\).

### Step 2: Identify the Highest Powers of Each Prime Factor
Now, we will take the highest power of each prime factor from the factorizations:

- For \(2\): The highest power is \(2^4\) (from 112).
- For \(3\): The highest power is \(3^2\) (from both 72 and 90).
- For \(5\): The highest power is \(5^1\) (from 90).
- For \(7\): The highest power is \(7^1\) (from 112).

### Step 3: Calculate the LCM
Now we can calculate the LCM by multiplying these highest powers together:

\[ \text{LCM} = 2^4 \times 3^2 \times 5^1 \times 7^1 \]

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:

\[ \text{LCM} = 16 \times 9 \times 5 \times 7 \]

Calculating it in parts:

- \(16 \times 9 = 144\)
- \(144 \times 5 = 720\)
- \(720 \times 7 = 5040\)

Thus, the LCM of 112, 72, and 90 is **5040**.

### Final Answer
The LCM of 112, 72, and 90 is **5040**.

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