Division of integer square by 4 leaves remainder 0 or 1

[bluemoon2188]

Hi,

I am looking for an explanation, if any, on why every integer square leaves remainder 0 or 1 on division by 4.

Appreciate your time and help

bluemoon2188

 

[dalcde]

I guess we should divide the integers into (1) even numbers (2) odd number.

Case 1:
$$n=2x$$
Then
$$n^2\equiv4x^2\equiv0 \mbox{ mod } 4$$

Case 2:
$$n=2x+1$$
Then
$$n^2\equiv4x^2+4x+1\equiv4(x^2+x)+1\equiv1 \mbox{ mod } 4$$

So in general, any even number squared equals 0 mod 4 and every odd number squared equals 1 mod 4. Hope that helps!

 

[HallsofIvy]

Actually, you can say more. Every odd integer, squared, has remainder 1 when divided by 4, every even integer, squared, is a multiple of 4.

Every integer is either even or odd. That is every integer is equal to 2n, for some integer n, or 2n+1 for some integer n.

(2n)²= 4n²

(2n+ 1)²= 4n²+ 4n+ 1

Two minutes too slow!

 

[bluemoon2188]

hey guys,

Thanks for the help. Didn't see that coming.

Cheers