Question

Show that \(a^{2}+b^{2}=0,1\), or 2 modulo 4 for all \(a, b \in Z\).

Short answer

The sum of two squares modulo 4 can be 0, 1, or 2.

Step-by-step solution

Understand Modulo 4

The modulus 4 calculation involves the possible remainders when a number is divided by 4. The remainders can be 0, 1, 2, or 3, which we use for simplifying expressions in our proof.

Determine Possible Values for Squares

Find the possible values for squares of integers modulo 4. Consider that any integer can be in one of four forms, depending on its remainder when divided by 4: 0, 1, 2, or 3. Calculate each case: - If an integer is 0 modulo 4, its square is 0 squared, resulting in 0 modulo 4. - If an integer is 1 modulo 4, its square is 1 squared, resulting in 1 modulo 4. - If an integer is 2 modulo 4, its square is 4, which is 0 modulo 4. - If an integer is 3 modulo 4, its square is 9, which is 1 modulo 4. Thus, the squares of integers modulo 4 can be either 0 or 1.

Analyze the Sum of Two Squares

Consider the expression \(a^2 + b^2\). Since the possible values for each square are 0 or 1 (from Step 2), analyze the possible outcomes:- Both \(a^2\) and \(b^2\) are 0 → \(a^2 + b^2 = 0+0 = 0\).- One is 0 and the other is 1 → \(a^2 + b^2 = 0+1 = 1\) or \(1+0 = 1\).- Both are 1 → \(a^2 + b^2 = 1+1 = 2\).Hence, the possible results for \(a^2 + b^2\) modulo 4 are 0, 1, or 2.

Key concepts

Integer Squares

When we talk about integer squares, we're looking at squaring whole numbers (integers). This concept is crucial in many branches of mathematics. Squaring an integer means multiplying it by itself. For any integer \(a\), squaring gives us \(a^2\). Understanding how these squares behave under various conditions, such as modular arithmetic, helps simplify many mathematical problems.
In the case of modulo 4, we are particularly interested in how these integer squares behave when divided by 4. We want to know what the remainder is when this squared value is divided by 4. This is an important step for solving problems that involve finding specific patterns or rules within number sets.

Modulo 4

Modulo arithmetic, often referred to as "clock arithmetic," deals with remainders. Specifically, modulo 4 means we're interested in how a number behaves when divided by 4. Every integer, when divided by 4, will leave a remainder of either 0, 1, 2, or 3.
When applying modulo 4 to integer squares:

• An integer with a remainder of 0 when divided by 4 will have its square also give a remainder of 0.
• An integer that leaves a remainder of 1 when divided by 4 will have a square remainder of 1.
• If the remainder is 2, the square will have a remainder of 0. This is because 2 squared is 4, and 4 divided by 4 leaves a remainder of 0.
• An integer with a remainder of 3 will also produce a square with a remainder of 1, as 3 squared is 9, and 9 divided by 4 gives a remainder of 1.

This simplification helps understand how integer squares behave under modulo 4, showing they can only result in 0 or 1 when squared.

Remainders

Remainders are a fundamental concept in understanding modulo arithmetic. They represent what is "left over" after division. In our exercise, understanding how remainders work when integer squares are divided by 4 becomes crucial.
For example, if we consider two squared integers, \(a^2\) and \(b^2\), we can predict their combined results based on their remainders:

• If both \(a^2\) and \(b^2\) leave a remainder of 0, their sum (0 + 0) leaves a remainder of 0.
• If one leaves a remainder of 0 and the other 1, their sum (0 + 1 or 1 + 0) leaves a remainder of 1.
• Finally, if both leave a remainder of 1, their sum (1 + 1) leaves a remainder of 2.

This shows that for the sum \(a^2 + b^2\) under modulo 4, the possible remainders are 0, 1, or 2. Recognizing these patterns is essential in many areas of mathematics and problem-solving, enabling clearer solutions and insights.