Question

prove that square of any integer leaves the remainder either 0 or 1 when divided by 4​

Answer 1

Every integer when squared leaves a the remainder 0 or 1 when divided by 4.

Here, we have two kinds of integers:

Even integers,

Odd integers,

An even integer can be expressed in the form 2n

And an odd integer can be expressed in the form 2n + 1

So, square of an even integer is $(2n)^{2}$ = $4n^{2}$

Square of an odd integer is $(2n + 1)^{2}$ = $4n^{2}$ + 4n + 1

$4n^{2}$ = 0 mod 4

4($n^{2}$ + n) + 1 = 1 mod 4

So, every even integer squared leaves a remainder 0 when divided by 4.

And every odd integer squared leaves a remainder 1 when divided by 4.

Hence, any integer squared leaves a reminder 0 or 1 when divided by 4.

Answer 2

Step-by-step explanation:

Hi there the answer is

Every integer when square leaves the reminder 0 or 1 when divided by 4.

Even integers,

Odd integers,

An even integer can be expressed in the form 2n

And an