A square with 1,536 soldiers are not possible, i.e. the square root of 1,536 is about 39.191, which means that 1,536 is not a square a number. In order for it to be a square number, we have to round up 39.191 (because it says "how many more soldiers") to 40 and 40×40=1,600 and 1,600-1,536, so we need 64 more soldiers.
The greatest 4 digit number is simple. If the square root of 10,000 is 100, than, one can assume that because 100-1=99, the resulting answer of 99×99 is less than 10,000, meaning that is is 4 digits. 99×99=9,801.