# Division of integer square by 4 leaves remainder 0 or 1

• bluemoon2188
In summary, the conversation discusses why every integer squared leaves a remainder of 0 or 1 when divided by 4. The explanation is that even numbers squared equal 0 mod 4 while odd numbers squared equal 1 mod 4. It is also mentioned that every integer is either even or odd and can be expressed as 2n or 2n+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.

bluemoon2188

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!

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!

hey guys,

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

Cheers

,

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.

## 1. What does it mean to divide an integer square by 4 and have a remainder of 0 or 1?

When dividing an integer square by 4 and the remainder is 0 or 1, it means that when the square root of the integer is divided by 4, there will be either a remainder of 0 or 1. For example, if we have the integer 16, its square root is 4, and when divided by 4, there is a remainder of 0. However, if we have the integer 15, its square root is not a whole number, but if we round it down to 3, when divided by 4, there will be a remainder of 3, which is equivalent to a remainder of 1.

## 2. Why is it important to consider the remainder when dividing an integer square by 4?

The remainder when dividing an integer square by 4 is important because it can provide useful information about the number itself. For example, if the remainder is 0, it means that the number is a perfect square and can be evenly divided by 4. If the remainder is 1, it means that the number is not a perfect square but its square root can be divided by 4 with a remainder of 1. This can also be helpful in solving certain mathematical problems and equations.

## 3. Can the remainder be any number other than 0 or 1 when dividing an integer square by 4?

No, the remainder when dividing an integer square by 4 can only be 0 or 1. This is because when dividing by 4, the possible remainders are 0, 1, 2, or 3. However, when dealing with integer squares, the only possible square roots are whole numbers, which means the remainder will always be either 0 or 1.

## 4. How can we determine the remainder when dividing an integer square by 4?

To determine the remainder when dividing an integer square by 4, we can use the modulus operator (%). This operator returns the remainder of a division operation. So, if we divide the square root of the integer by 4 and use the modulus operator, the result will be either 0 or 1, indicating the remainder.

## 5. What are some real-life applications of dividing an integer square by 4 and considering the remainder?

One real-life application of dividing an integer square by 4 and considering the remainder is in the field of computer science and coding. The modulus operator is often used to determine if a number is even or odd, which can be helpful in writing efficient code. Additionally, the remainder can also be used in cryptography and encoding data. In mathematics, the remainder can also be used in various algebraic equations and problem-solving.

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