Problem 12

Question

Prove each directly. The square of every integer of the form $3 k+1$ is also of the same form, where $k$ is an arbitrary integer.

Step-by-Step Solution

Verified

Answer

Define the given integer $n = 3k + 1$, where $k$ is an arbitrary integer. Square the integer $n$, then expand and simplify: $n^2 = 9k^2 + 6k + 1$. Factor out a 3 to rewrite the expression in the same form: $n^2 = 3(3k^2 + 2k) + 1$. Let $m = 3k^2 + 2k$, which is also an integer, so $n^2 = 3m + 1$. Thus, the square of every integer of the form $3k + 1$ is also of the same form.

1Step 1: Define the given expression and square it

Let $n$ be an integer of the form $3k + 1$, where $k$ is an arbitrary integer. Therefore, we have: \[n = 3k + 1\] Now, let's square the integer $n$: \[n^2 = (3k + 1)^2\]

2Step 2: Expand and simplify

Next, we expand and simplify this expression: \[n^2 = (3k + 1)(3k + 1)\] \[n^2 = 9k^2 + 6k + 1\]

3Step 3: Rewrite the expression in the same form

Now, let's try to rewrite the expression in the same form as the original integer $n$, which had the form $3k + 1$. We can achieve this by factoring out a 3: \[n^2 = 3(3k^2 + 2k) + 1\]

4Step 4: Define a new integer

Let $m = 3k^2 + 2k$; note that $m$ is also an integer since $k$ is an integer and the sum of two integers is also an integer. Therefore, we can rewrite the expression as: \[n^2 = 3m + 1\]

5Step 5: Conclusion

We have shown that when we square an integer $n$ of the form $3k + 1$, the resulting expression is also of the form $3m + 1$, where $m$ is a different arbitrary integer. Therefore, the statement is proven: the square of every integer of the form $3k + 1$ is also of the same form.

Key Concepts

Integer SquaresMathematical ProofsAlgebraic Manipulation

Integer Squares

When we discuss integer squares, we mean the process of taking an integer and multiplying it by itself. A square is a type of power where the exponent is 2. The square of an integer is always non-negative.

For example, the square of 3 is calculated as: $3^2 = 3 \times 3 = 9$.
Similarly, the square of -3 is also 9, since $(-3)^2 = (-3) \times (-3) = 9$.

In our original exercise, we square an integer of the form $3k + 1$. This means looking at how substituting and expanding these expressions helps us delve into specific number set properties, such as in quadratic expansion or modular arithmetic. Squaring plays a significant role in these contexts, making it a prevalent concept in algebra and number theory.

Mathematical Proofs

Mathematical proofs are structured attempts to verify a statement's truth beyond doubt. Proofs can take many forms, but they all rely on logical reasoning and existing mathematical laws. The direct proof, used in this exercise, builds from known facts to arrive at a conclusion.

Direct Proof: This type of proof entails assuming a fact that needs to be proven and then logically deriving the desired result. It is a straightforward method and can be very intuitive.
Proofs are crucial in mathematics as they help confirm theorems and ensure that mathematical laws hold under various conditions.

In our exercise example, we directly proved that squaring an integer of a specific form yields another integer of similar form. This involves defining our entities, performing algebraic manipulation, and logically arranging the information to arrive at the proof.

Algebraic Manipulation

Algebraic manipulation is the art of rewriting mathematical expressions using rules and operations of algebra. It can simplify problems and give us insights into the structure of expressions.

Expanding Expressions: When we expanded $(3k + 1)^2$, it included multiplying terms and combining like terms to simplify the expression to $9k^2 + 6k + 1$.
Simplifying Terms: Simplification involves combining like terms and reducing expressions to their simplest form, often by factoring out common elements, as we did by rewriting the expression as $3m + 1$.

Through algebraic manipulation, we're able to transition from an individual term ($3k + 1$), to its square, and finally express it back into the desired format ($3m + 1$). Algebraic manipulation helps reveal structural patterns within numbers and expressions, aiding in many proof contexts like the one above.

Problem 12

Problem 13

Other exercises in this chapter

Problem 12

Let $\mathrm{P}(x): x^{2} > x, \mathrm{Q}(x): x^{2}=x,$ and the UD $=$ set of integers. Determine the truth value of each proposition. $$(\forall x)|\mathrm

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Problem 12

Verify each, where $f$ denotes a contradiction. $$p \rightarrow(q \vee r) \equiv(p \wedge \sim q) \rightarrow r$$

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Problem 13

Give the simplest possible conclusion in each argument. Assume each premise is true. $$\begin{aligned} &p \leftrightarrow q\\\ &\sim p \vee r\\\ &\sim r \end{al

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Problem 13

Prove each directly. The square of every integer of the form $4 k+1$ is also of the same form, where $k$ is an arbitrary integer.

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