Question

Show that the square of 2q+1 is infact the form 4s+1 and thus prove that every integer square leaves remainder 0 or 1 on division by 4.

Answer 1

1) The square of (2q+1) is in the form of (4s+1)

Since q and q^2 will belong to set of integers, hence we can write

so we can say that

The integer will belong to two form, either the integers will be even or odd

Even integer: a = 2p, where p is an integer

Since p is an integer, hence p^2 will also be an integer

So the a^2 is divisible by 4, hence it will leave the remainder 0

Odd integer: a = 2p+1, where p is an integer

Since p is an integer, hence p^2 will also be an integer

So the a^2 4s_2 will be divisible by 4, hence it will leave the remainder 1 after division by 4