How many numbers between 400 and 800 are divisible by 4, 5 and 6?

Options

A. 7
B. 8
C. 9
D. 10

Answer

To solve the question of how many numbers between 400 and 800 are divisible by 4, 5, and 6, we can follow these steps:

### Step 1: Find the LCM of 4, 5, and 6
To determine the numbers that are divisible by 4, 5, and 6, we first need to find the least common multiple (LCM) of these numbers.

- The prime factorization of 4 is \(2^2\).
- The prime factorization of 5 is \(5^1\).
- The prime factorization of 6 is \(2^1 \times 3^1\).

The LCM is found by taking the highest power of each prime factor:
- For 2, the highest power is \(2^2\).
- For 3, the highest power is \(3^1\).
- For 5, the highest power is \(5^1\).

Thus, the LCM is:
\[ LCM = 2^2 \times 3^1 \times 5^1 = 4 \times 3 \times 5 = 60 \]

### Step 2: Count the multiples of 60 between 400 and 800
Next, we need to find how many multiples of 60 exist between 400 and 800.

#### Finding multiples of 60 up to 800:
To find the largest multiple of 60 that is less than or equal to 800, we divide 800 by 60:
\[ \frac{800}{60} \approx 13.33 \]
The largest integer is 13, so the largest multiple of 60 is:
\[ 60 \times 13 = 780 \]

#### Finding multiples of 60 up to 400:
Now, we find how many multiples of 60 are less than or equal to 400:
\[ \frac{400}{60} \approx 6.67 \]
The largest integer is 6, so the largest multiple of 60 is:
\[ 60 \times 6 = 360 \]

### Step 3: Calculate the number of multiples between 400 and 800
To find the count of multiples of 60 between 400 and 800, we subtract the count of multiples up to 400 from the count of multiples up to 800:
\[ \text{Count from 1 to 800} - \text{Count from 1 to 400} = 13 - 6 = 7 \]

### Conclusion
Thus, the total number of numbers between 400 and 800 that are divisible by 4, 5, and 6 is **7**.

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