Question

How can I prove that any perfect square divided by 4 gives remainders 1 and 0 only?

Answer 1

Any number which on division by 4 leaves a remainder 2 or 3 is not a perfect square.

We should divide the integers into even numbers and odd numbers.

Case 1 (even numbers):

$n=2*x$

Then:

$n^2 ≡ 4*x^2 ≡ 0 mod 4$

(because if you will divide it by 4 it will be 0)

Case 2 (odd numbers):

$n = 2*x + 1$

 Then:

$n^2 ≡ 4*x^2 + 4*x + 1 ≡ 4*( x2 + x ) + 1 ≡ 1 mod 4$

(because if you will divide it without "+1" by 4 it will be 0, with "+1" remainder will be 1)

Thus, any even number squared equals 0 mod 4 and every odd number squared equals 1 mod 4. Any number which on division by 4 leaves a remainder 2 or 3 is not a perfect square.