Multiples of 7 and 11

Since 7 and 11 are coprime, the answer must be of the form $7 \times 11 \times N$ , where $N$ is a positive integer. Since the answer is a 3-digit number, we have $7 \times 11 \times N < 1000$ . This implies $N < 12.9\ldots$ . Hence, the largest 3-digit integer which is a multiple of both 7 and 11 is $7\times 11 \times 12 = 924$ .

Note: It is useful to remember that $1001 = 7 \times 11 \times 13$ .

Since the answer must be divisible by $7$ and $11$ (which are coprime) then ans. has to be divisible by $77$ .The other factor beside $77$ should be less than $13\;(7\times11\times13=1001)$ so the other factor will be $12$ so the number is $77\times12=\boxed{924}$