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

 

[CellCoree]

,

Thank you for your question. The reason why every integer square leaves a remainder of 0 or 1 when divided by 4 is due to the properties of even and odd numbers.

An integer can be classified as either even or odd. An even number is any number that is divisible by 2, while an odd number is any number that is not divisible by 2.

When we square an even number, the result will always be divisible by 4. For example, 2 squared is 4, which is divisible by 4. 4 squared is 16, which is also divisible by 4. This is because when we multiply an even number by itself, we are essentially multiplying it by 2 twice. So, the result will always be divisible by 4.

On the other hand, when we square an odd number, the result will always leave a remainder of 1 when divided by 4. For example, 3 squared is 9, which leaves a remainder of 1 when divided by 4. 5 squared is 25, which also leaves a remainder of 1 when divided by 4. This is because when we multiply an odd number by itself, we are essentially multiplying it by 2 once and adding 1. So, the result will always leave a remainder of 1 when divided by 4.

Therefore, when we divide any integer square by 4, the remainder will always be either 0 or 1, depending on whether the original number was even or odd. This is a fundamental property of numbers and can be seen in various mathematical concepts and applications. I hope this explanation helps to clarify the reason behind this phenomenon.