The circumferences of four wheels are `"50 cm, 60 cm, 75 cm, and 100 cm"`, . They start moving simultaneously. What least distance should they cover so that each wheel makes a complete number of revolutions ?

Options

A. `3.5 m `
B. `3 m `
C. `4m `
D. `4.5m `

Answer

To find the least distance that the four wheels should cover so that each wheel makes a complete number of revolutions, we need to calculate the Least Common Multiple (LCM) of their circumferences. Here’s a step-by-step solution:

### Step 1: Identify the circumferences of the wheels
The circumferences of the wheels are:
- Wheel 1: 50 cm
- Wheel 2: 60 cm
- Wheel 3: 75 cm
- Wheel 4: 100 cm

### Step 2: Find the prime factorization of each circumference
- **50 cm**:
- \(50 = 2 \times 5^2\)

- **60 cm**:
- \(60 = 2^2 \times 3 \times 5\)

- **75 cm**:
- \(75 = 3 \times 5^2\)

- **100 cm**:
- \(100 = 2^2 \times 5^2\)

### Step 3: Determine the highest power of each prime factor
Now, we will take the highest power of each prime factor from the factorizations:
- For **2**: The highest power is \(2^2\) (from 60 and 100).
- For **3**: The highest power is \(3^1\) (from 60 and 75).
- For **5**: The highest power is \(5^2\) (from 50, 75, and 100).

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

Calculating this step-by-step:
1. \(2^2 = 4\)
2. \(3^1 = 3\)
3. \(5^2 = 25\)

Now multiply them together:
\[ \text{LCM} = 4 \times 3 \times 25 \]

Calculating:
1. \(4 \times 3 = 12\)
2. \(12 \times 25 = 300\)

### Final Answer
The least distance that the wheels should cover so that each wheel makes a complete number of revolutions is **300 cm**.

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