Find the greatest three digit number and the smallest four-digit which is exactly divisible by the numbers 84 and 112.

We know, 84 = 2 $\times$ 2 $\times$ 3 $\times$ 7
and, 112 = 2 $\times$ 2 $\times$ 2 $\times$ 2 $\times$ 7
LCM(84, 112) = 2 $\times$ 2 $\times$ 2 $\times$ 2 $\times$ 3 $\times$ 7 = 336

Every integral multiple of the LCM of two given numbers is divisible by the given numbers.
Let the greatest three digit number be 336 $\times$ #.
For the greatest 3-digits number,
336 $\times$ # < 1000
=> # < 3
=> # = 2

The greatest three digit number be:
336 $\times$ # = 336 $\times$ 2 = 672

If # = 3, the multiple would have become 1008, which in that case would have been the smallest 4 digit number divisible by 84 and 112.