This again is a question that we need to solve by trial and error. Clearly, N is an odd number. So, the remainder when we divide N by 24 has to be odd.
If the remainder when we divide N by 24 = 1, then N^{2} also has a remainder of 1. we can also see that if the remainder when we divide N by 24 is -1, then N^{2} a remainder of 1.
When remainder when we divide N by 24 is ±3, then N^{2} has a remainder of 9.
When remainder when we divide N by 24 is ±5, then N^{2} has a remainder of 1.
When remainder when we divide N by 24 is ±7, then N^{2} has a remainder of 1.
When remainder when we divide N by 24 is ±9, then N^{2} has a remainder of 9.
When remainder when we divide N by 24 is ±11, then N^{2} has a remainder of 1.
So, the remainder when we divide N by 24 could be ±1, ±5, ±7 or ±11.
Or, the possible remainders when we divide N by 24 are 1, 5, 7, 11, 13, 17, 19, 23.
Or, the possible remainders when we divide N by 12 are 1, 5, 7, 11.
Correct Answer: 1, 5, 7, 11