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

The correct option is 1008.

To find 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 three numbers first.

Step 1: Find the prime factorization of each number:

6 = 2 × 3
7 = 7
4 = 2²

Step 2: Calculate the LCM of 6, 7, and 4:

The LCM is obtained by taking the highest power of each prime factor involved:

$\text{LCM}(6,7,4)=2^2\times 3\times 7=4\times 3\times 7=84$

Any common multiple of 6, 7, and 4 must also be a multiple of 84.

Step 3: Find the smallest four-digit multiple of 84:

The smallest four-digit number is 1000.

Now, divide 1000 by 84 to find the remainder:

$1000÷84=11\text{with a remainder of}76$

Since 84 × 11 = 924 (a three-digit number), the next multiple will be the smallest four-digit number:

$84\times 12=1008$

Alternatively, we can add the difference between the divisor and the remainder to 1000:

$1000+(84-76)=1000+8=1008$

Thus, the smallest four-digit number which is a multiple of 6, 7, and 4 is 1008.