How many numbers between 200 and 600 are exactly divisible by 4 ,5 and 6?

Options

A. 5
B. 16
C. 10
D. 6

Answer

To find how many numbers between 200 and 600 are exactly divisible by 4, 5, and 6, we will follow these steps:

### Step 1: Find the LCM of 4, 5, and 6
To find the LCM (Least Common Multiple), we first break down each number into its prime factors:

- \(4 = 2^2\)
- \(5 = 5^1\)
- \(6 = 2^1 \times 3^1\)

Now, we take the highest power of each prime factor:

- From \(4\), we take \(2^2\)
- From \(5\), we take \(5^1\)
- From \(6\), we take \(3^1\)

Now, we multiply these together to find the LCM:

\[ \text{LCM} = 2^2 \times 3^1 \times 5^1 = 4 \times 3 \times 5 = 60 \]

### Step 2: Identify the range of numbers between 200 and 600 that are divisible by 60
Next, we need to find the multiples of 60 that lie between 200 and 600.

To do this, we can find the smallest multiple of 60 that is greater than or equal to 200 and the largest multiple of 60 that is less than or equal to 600.

- **Smallest multiple of 60 greater than or equal to 200**:
- Divide 200 by 60: \( \frac{200}{60} \approx 3.33 \)
- The smallest integer greater than 3.33 is 4.
- Thus, the smallest multiple is \( 60 \times 4 = 240 \).

- **Largest multiple of 60 less than or equal to 600**:
- Divide 600 by 60: \( \frac{600}{60} = 10 \)
- Thus, the largest multiple is \( 60 \times 10 = 600 \).

### Step 3: List all multiples of 60 between 240 and 600
Now we can list the multiples of 60 from 240 to 600:

- \(240\)
- \(300\)
- \(360\)
- \(420\)
- \(480\)
- \(540\)
- \(600\)

### Step 4: Count the numbers
Now we count the multiples listed:

- \(240, 300, 360, 420, 480, 540, 600\)

There are **7 numbers** in total.

### Final Answer
Thus, the total number of numbers between 200 and 600 that are exactly divisible by 4, 5, and 6 is **7**.

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