Many of you came back with the correct answer this time – thanks. I got the first correct from Shreya Mantripragada (9 year old, Bay Area). Also got correct answer from Anubhav Garg who explained his answer really well, Shruti Mittal and Anisha Sharma Goyal, though it took some pushing from my side!
The correct answer is 101 as many of you pointed out. Here is the logic:
From rule #1 (number -1 is a perfect square) and #3 (number -1 is divisible by 5) one can conclude that the number must be divisible by 25.
From rule #4, we know that number-5 is divisible by 4 (since it is divisible by 8) and therefore number -1 will also be divisible by 4.
From the above two conclusions, we can conclude that number -1 needs to be divisible by 25 as well as 4 (both of which are squares as well), therefore the smallest such ( number -1) can be 100, and therefore the smallest number can be 101. Quick inspection will indicate that all the conditions are satisfied and therefore 101 is the smallest such answer.
Hope you all enjoyed the puzzle!