The smallest four-digit number which is a multiple of 6, 7 and 4 is :

Question

Accepted Answer

To determine the smallest four-digit number that is a multiple of 6, 7, and 4, we need to find the least common multiple (LCM) of these numbers and ensure that it is a four-digit number.

1. **Finding the LCM of 6, 7, and 4:**
   - The prime factorization of 6 is $2 \times 3$.
   - The prime factorization of 7 is $7$.
   - The prime factorization of 4 is $2^2$.

To find the LCM, we take the highest power of each prime number that appears in the factorizations:
   - The highest power of 2 is $2^2$.
   - The highest power of 3 is $3$.
   - The highest power of 7 is $7$.

Therefore, the LCM is $2^2 \times 3 \times 7 = 4 \times 3 \times 7 = 84$.

2. **Finding the smallest four-digit number that is a multiple of 84:**
   - The smallest four-digit number is 1000.
   - To find the smallest four-digit multiple of 84, we divide 1000 by 84 and round up to the nearest whole number:
     $$
     \frac{1000}{84} \approx 11.90
     $$
   - Rounding up gives us 12. Therefore, the smallest integer greater than 11.90 is 12.
   - Multiply 84 by 12 to find the smallest four-digit multiple:
     $$
     84 \times 12 = 1008
     $$

3. **Verification of Option 2 (1008):**
   - 1008 is divisible by 6, 7, and 4, as it is a multiple of their LCM, 84.

4. **Explanation of why other options are incorrect:**
   - **Option 1 (1006):** 1006 is not divisible by 84. Dividing 1006 by 84 does not yield a whole number.
   - **Option 3 (1000):** 1000 is not divisible by 84. Dividing 1000 by 84 does not yield a whole number.
   - **Option 4 (1002):** 1002 is not divisible by 84. Dividing 1002 by 84 does not yield a whole number.

Therefore, the correct answer is Option 2 (1008), as it is the smallest four-digit number that is a multiple of 6, 7, and 4.