A number that is divisible by all the numbers from 1 to 10 (both inclusive) is

Question

Accepted Answer

The correct answer to the question is Option 4: 2520. Let's explore why this is the case and why the other options are incorrect.

### Explanation for Option 4: 2520

To determine a number that is divisible by all the numbers from 1 to 10, we need to find the least common multiple (LCM) of these numbers. The LCM is the smallest number that is a multiple of each of the numbers in the set.

1. **Prime Factorization**: 
   - 1 is not needed for LCM as it does not affect multiplication.
   - 2 is a prime number.
   - 3 is a prime number.
   - 4 is $2^2$.
   - 5 is a prime number.
   - 6 is $2 \times 3$.
   - 7 is a prime number.
   - 8 is $2^3$.
   - 9 is $3^2$.
   - 10 is $2 \times 5$.

2. **LCM Calculation**:
   - For 2, the highest power is $2^3$ (from 8).
   - For 3, the highest power is $3^2$ (from 9).
   - For 5, the highest power is $5^1$ (from 5 or 10).
   - For 7, the highest power is $7^1$ (from 7).

Therefore, the LCM is $2^3 \times 3^2 \times 5^1 \times 7^1 = 8 \times 9 \times 5 \times 7 = 2520$.

Thus, 2520 is the smallest number that is divisible by all numbers from 1 to 10.

### Explanation for Why Other Options are Incorrect

1. **Option 1: 10**
   - 10 is divisible by 1, 2, 5, and 10, but not by 3, 4, 6, 7, 8, or 9. Therefore, it cannot be the correct answer.

2. **Option 2: 100**
   - 100 is divisible by 1, 2, 4, 5, and 10, but not by 3, 6, 7, 8, or 9. Hence, it is not the correct answer.

3. **Option 3: 604**
   - 604 is divisible by 1, 2, and 4, but not by 3, 5, 6, 7, 8, 9, or 10. Therefore, it is not the correct answer.

In conclusion, Option 4: 2520 is the only number that meets the criteria of being divisible by all numbers from 1 to 10, making it the correct answer.