How many numbers from 1 to 1000 are divisible by 2, 3, 4 and 5 ?

Options

A. 16
B. 17
C. 32
D. None of the above

Answer

To find how many numbers from 1 to 1000 are divisible by 2, 3, 4, and 5, we can follow these steps:

### Step 1: Find the LCM of the numbers 2, 3, 4, and 5
To determine how many numbers are divisible by all four numbers, we first need to find their least common multiple (LCM).

- The prime factorization of each number is:
- 2 = 2
- 3 = 3
- 4 = 2^2
- 5 = 5

- To find the LCM, we take the highest power of each prime number:
- The highest power of 2 is 2^2 (from 4)
- The highest power of 3 is 3^1 (from 3)
- The highest power of 5 is 5^1 (from 5)

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

### Step 2: Count how many multiples of the LCM are there from 1 to 1000
Next, we need to find how many multiples of 60 are there from 1 to 1000.

- We can find this by dividing 1000 by 60:
\[ \text{Number of multiples} = \left\lfloor \frac{1000}{60} \right\rfloor \]

Calculating this gives:
\[ \frac{1000}{60} \approx 16.67 \]

Taking the floor value, we find:
\[ \left\lfloor 16.67 \right\rfloor = 16 \]

### Step 3: Conclusion
Therefore, the total number of integers from 1 to 1000 that are divisible by 2, 3, 4, and 5 is **16**.

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