1.) Prove that a square of some integer has to leave a remainder of either 0 or 1 when it is divided by 4.
2.) Prove that there is no number in the following sequence: 11, 111, 1111, 11111, ... that's a perfect square.
1. Let the number you're after be p. There are only 4 possible senarios in modulo 4.
p = 0 (mod 4) => p^2 = 0 (mod 4)
p = 1 (mod 4) => p^2 = 1 (mod 4)
p = 2 (mod 4) => p^2 = 4 = 0 (mod 4)
p = 3 (mod 4) => p^2 = 9 = 1 (mod 4)
Therefore any square number would have the property that p = 0 or 1 (mod 4)
2. As you can see, you can write each of the 1111.....11 (n digits) numbers as:
(n-2) digits ---> 111...111 x 100 + 11
In mod 4, this is the same as 111...111 x (4 x 25) + 11 = 11 = 3 (mod 4)
Since 111...111 is not equal to 0 or 1 mod 4, it cannot be a perfect square (from 1)