Question

Show that the square of any integer is of the form \(3 m\) or \(3 m+1\) but not of the form \(3 m+2\).

Short answer

Question: Prove that the square of any integer is of the form \(3m\) or \(3m+1\), but not of the form \(3m+2\). Short Answer: We first analyzed the possible remainders when an integer is divided by 3, which are 0, 1, and 2. For each case, we squared the integer and found that the square is either of the form \(3m\) or \(3m+1\), with no squares in the form of \(3m+2\). Consequently, we proved that the square of any integer is of the form \(3m\) or \(3m+1\), but not of the form \(3m+2\).

Step-by-step solution

Identify possible remainders

When dividing an integer by 3, there can be 3 possible remainders: 0, 1, or 2.

Analyse the case when the remainder is 0

Let's consider an integer \(n\) such that its remainder when divided by 3 is 0. Then, \(n = 3k\) for some integer \(k\). When we square \(n\), we have: \(n^2 = (3k)^2 = 9k^2 = 3(3k^2)\). Since \(3k^2\) is an integer, the square of \(n\) is of the form \(3m\) with \(m = 3k^2\).

Analyse the case when the remainder is 1

Let's consider an integer \(n\) such that its remainder when divided by 3 is 1. Then, \(n = 3k + 1\) for some integer \(k\). When we square \(n\), we have: \(n^2 = (3k + 1)^2 = 9k^2 + 6k + 1 = 3(3k^2 + 2k) + 1\). Since \(3k^2 + 2k\) is an integer, the square of \(n\) is of the form \(3m+1\) with \(m = 3k^2 + 2k\).

Analyse the case when the remainder is 2

Let's consider an integer \(n\) such that its remainder when divided by 3 is 2. Then, \(n = 3k + 2\) for some integer \(k\). When we square \(n\), we have: \(n^2 = (3k + 2)^2 = 9k^2 + 12k + 4 = 3(3k^2 + 4k) + 1\). Since \(3k^2 + 4k\) is an integer, the square of \(n\) is of the form \(3m+1\) with \(m = 3k^2 + 4k\).

Conclude by comparing the results

We analyzed all possible cases of integer remainders when divided by 3. In each case, we found that the square is either of the form \(3m\) or \(3m+1\). We never had a square in the form of \(3m+2\). Therefore, we can conclude that the square of any integer is of the form \(3m\) or \(3m+1\), but not of the form \(3m+2\).

Key concepts

Understanding Integers

Integers are simply the whole numbers that include positive numbers, negative numbers, and zero. These numbers do not have fractional or decimal parts.
They are essential in counting, ordering, and many mathematical operations.
Some properties of integers include:

• Closure: The sum or product of any two integers is always an integer.
• Associativity: For addition and multiplication, changing the grouping of integers does not change the result.
• Commutativity: The order in which you add or multiply integers does not change the result.

In our exercise, every integer can be expressed in a special form when divided by a certain number; in this case, 3.
This property is fundamental in finding patterns such as in the discovery of the forms of squared integers.

Discovering Remainders

When dividing a number, the remainder is what's left over after division.
For example, if you divide 10 by 3, the quotient is 3 and the remainder is 1, because 10 can be expressed as 3 times 3 plus 1.
Remainders are crucial in modular arithmetic, which is a central concept in this exercise.
Here's how remainders work:

• When an integer is divided by 3, the possible remainders are 0, 1, or 2.
• These remainders represent the values left when an integer isn’t a perfect multiple of 3.

Understanding remainders helps in determining different forms of squared numbers. In the example, for remainders 0 and 1, we see particular patterns aligning with forms \(3m\) and \(3m+1\). The concept of remainders shows us why some forms, like \(3m+2\), aren't possible for squared integers.

Exploring Modulo Arithmetic

Modulo arithmetic, often just called "mod," is a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value, known as the modulus.
In this exercise, we're using modulo 3 to determine the form of squared integers.
The basics of modulo arithmetic include:

• \( a \mod n \) gives the remainder when \( a \) is divided by \( n \).
• This type of arithmetic is fundamental in various fields like computer science, cryptography, and number theory.
• It helps simplify complex arithmetic operations by focusing on the remainder.

Our problem specifically explores how squaring different integers translates in modulo 3.
For a number \( n \), if \( n \equiv 0, 1, \) or \( 2 \pmod{3} \), squaring it shows it’s either \( 3m \) or \( 3m+1 \), but not \( 3m+2 \).
This insight highlights the power and simplicity of using modulo arithmetic for analyzing numbers.