UPSC 2014 CDS I Math Paper

The remainder when 4 divides the square of an integer is either 0 or 1.
So, the sum of squares of two integers when divided by 4 leaves the remainder 0, 1 or 2.
Therefore, no integer of the form 4k + 3, where k is an integer, can be expressed as the sum of two squares.
Hence, statement 1 is correct.

Any odd number can be expressed in the form 2n + 1
(2n+1)2=4n2+4n+1
=4n(n+1)+1
Now, the product of any two consecutive integers is always divisible by 2.
∴(2n+1)2=4×2k+1=8k+1
where k is an integer
Hence, statement 2 is correct