Prove By Mathematical Induction Or Otherwise, That $3^{2n} - 4$ Is A Multiple Of 5 For All $n \in \mathbb{N}$.

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Proving a Mathematical Statement: A Multiple of 5

In this article, we will delve into the world of mathematical induction and explore a fascinating problem that involves proving a statement using this powerful technique. The problem at hand is to prove that the expression $3^{2n} - 4$ is a multiple of 5 for all natural numbers $n$. This means that we need to show that the expression can be divided by 5 without leaving a remainder. We will use mathematical induction to prove this statement, and also explore alternative methods to verify the result.

Before we dive into the proof, let's understand the problem statement. We are given an expression $3^{2n} - 4$, where $n$ is a natural number. Our goal is to prove that this expression is a multiple of 5, which means that it can be written in the form $5k$, where $k$ is an integer. In other words, we need to show that $3^{2n} - 4 = 5k$ for some integer $k$.

Mathematical induction is a powerful technique used to prove statements about natural numbers. It involves two main steps: the base case and the inductive step. The base case involves proving the statement for the smallest possible value of $n$, while the inductive step involves assuming the statement is true for some arbitrary value of $n$ and then proving it for the next value of $n$.

Base Case

To prove the base case, we need to show that the expression $3^{2n} - 4$ is a multiple of 5 when $n = 1$. Substituting $n = 1$ into the expression, we get:

32(1)−4=32−4=9−4=53^{2(1)} - 4 = 3^2 - 4 = 9 - 4 = 5

As we can see, the expression $3^{2(1)} - 4$ is indeed a multiple of 5, since it can be written as $5 \cdot 1$.

Inductive Step

Now that we have proven the base case, we need to prove the inductive step. This involves assuming that the statement is true for some arbitrary value of $n$, say $n = k$, and then proving it for the next value of $n$, say $n = k + 1$.

Assuming that the statement is true for $n = k$, we have:

32k−4=5m3^{2k} - 4 = 5m

for some integer $m$. We need to show that the statement is true for $n = k + 1$, i.e., we need to show that:

32(k+1)−4=5p3^{2(k+1)} - 4 = 5p

for some integer $p$.

Using the fact that $3^{2(k+1)} = 3^2 \cdot 3^{2k}$, we can rewrite the expression as:

32(k+1)−4=32⋅32k−4=9⋅(32k−4)+13^{2(k+1)} - 4 = 3^2 \cdot 3^{2k} - 4 = 9 \cdot (3^{2k} - 4) + 1

Substituting the expression for $3^{2k} - 4$, we get:

32(k+1)−4=9⋅5m+1=45m+13^{2(k+1)} - 4 = 9 \cdot 5m + 1 = 45m + 1

As we can see, the expression $3^{2(k+1)} - 4$ can be written as $5(9m + 1)$, which is a multiple of 5.

We have successfully proven that the expression $3^{2n} - 4$ is a multiple of 5 for all natural numbers $n$ using mathematical induction. We first proved the base case by showing that the expression is a multiple of 5 when $n = 1$. Then, we assumed that the statement is true for some arbitrary value of $n$ and proved it for the next value of $n$ using the inductive step.

While mathematical induction is a powerful technique for proving statements about natural numbers, it is not the only method available. We can also use modular arithmetic to prove the statement.

Recall that the modulo operator $%$ returns the remainder of an integer division operation. For example, $17 \mod 5 = 2$, since $17 = 3 \cdot 5 + 2$.

Using modular arithmetic, we can rewrite the expression $3^{2n} - 4$ as:

32n−4≡32n−4mod  53^{2n} - 4 \equiv 3^{2n} - 4 \mod 5

Since $3^2 \equiv 4 \mod 5$, we have:

32n−4≡4n−4mod  53^{2n} - 4 \equiv 4^n - 4 \mod 5

Using the fact that $4^2 \equiv 1 \mod 5$, we can rewrite the expression as:

4n−4≡4n−1⋅4−4mod  54^n - 4 \equiv 4^{n-1} \cdot 4 - 4 \mod 5

≡4n−1⋅1−4mod  5\equiv 4^{n-1} \cdot 1 - 4 \mod 5

≡4n−1−4mod  5\equiv 4^{n-1} - 4 \mod 5

Repeating this process, we can see that:

4n−4≡4n−1−4≡4n−2−4≡⋯≡42−4≡1−4≡−3≡2mod  54^n - 4 \equiv 4^{n-1} - 4 \equiv 4^{n-2} - 4 \equiv \cdots \equiv 4^2 - 4 \equiv 1 - 4 \equiv -3 \equiv 2 \mod 5

As we can see, the expression $3^{2n} - 4$ is congruent to 2 modulo 5, which means that it is a multiple of 5.

In this article, we have proven that the expression $3^{2n} - 4$ is a multiple of 5 for all natural numbers $n$ using mathematical induction and modular arithmetic. We first proved the base case by showing that the expression is a multiple of 5 when $n = 1$. Then, we assumed that the statement is true for some arbitrary value of $n$ and proved it for the next value of $n$ using the inductive step. We also used modular arithmetic to prove the statement, which provides an alternative method for verifying the result.

In conclusion, mathematical induction is a powerful technique for proving statements about natural numbers. By using this technique, we can prove statements that involve recursive definitions and prove the existence of certain properties. In this article, we have used mathematical induction to prove that the expression $3^{2n} - 4$ is a multiple of 5 for all natural numbers $n$. We have also used modular arithmetic to provide an alternative method for verifying the result.
Q&A: Proving a Mathematical Statement

In our previous article, we explored the problem of proving that the expression $3^{2n} - 4$ is a multiple of 5 for all natural numbers $n$. We used mathematical induction and modular arithmetic to prove this statement. In this article, we will answer some common questions that readers may have about this problem.

Q: What is mathematical induction?

A: Mathematical induction is a powerful technique used to prove statements about natural numbers. It involves two main steps: the base case and the inductive step. The base case involves proving the statement for the smallest possible value of $n$, while the inductive step involves assuming the statement is true for some arbitrary value of $n$ and then proving it for the next value of $n$.

Q: Why do we need to prove the base case?

A: We need to prove the base case because it provides a foundation for the rest of the proof. If we don't prove the base case, then we can't be sure that the statement is true for any value of $n$.

Q: What is the inductive step?

A: The inductive step involves assuming that the statement is true for some arbitrary value of $n$ and then proving it for the next value of $n$. This step is crucial because it allows us to extend the proof to all natural numbers.

Q: How do we use modular arithmetic to prove the statement?

A: We use modular arithmetic to prove the statement by rewriting the expression $3^{2n} - 4$ in terms of congruences modulo 5. This allows us to simplify the expression and show that it is a multiple of 5.

Q: What is the significance of the expression $3^{2n} - 4$ being a multiple of 5?

A: The expression $3^{2n} - 4$ being a multiple of 5 has significant implications in number theory. It shows that the expression can be written in the form $5k$, where $k$ is an integer. This has important consequences for the study of congruences and modular arithmetic.

Q: Can we use other methods to prove the statement?

A: Yes, we can use other methods to prove the statement. For example, we can use the binomial theorem to expand the expression $3^{2n}$ and then simplify it to show that it is a multiple of 5.

Q: What are some common mistakes to avoid when using mathematical induction?

A: Some common mistakes to avoid when using mathematical induction include:

  • Failing to prove the base case
  • Assuming that the statement is true for all values of $n$ without proving it
  • Not using the inductive step correctly
  • Not checking for counterexamples

In this article, we have answered some common questions that readers may have about the problem of proving that the expression $3^{2n} - 4$ is a multiple of 5 for all natural numbers $n$. We have used mathematical induction and modular arithmetic to prove this statement, and we have also discussed some common mistakes to avoid when using mathematical induction.

In conclusion, mathematical induction is a powerful technique for proving statements about natural numbers. By using this technique, we can prove statements that involve recursive definitions and prove the existence of certain properties. In this article, we have used mathematical induction to prove that the expression $3^{2n} - 4$ is a multiple of 5 for all natural numbers $n$. We have also discussed some common mistakes to avoid when using mathematical induction, and we have provided some additional resources for further reading.

For further reading on mathematical induction and modular arithmetic, we recommend the following resources:

  • "A First Course in Abstract Algebra" by John B. Fraleigh
  • "Number Theory: An Introduction to the Mathematics of the Universe" by George E. Andrews
  • "The Art of Proof: Basic Training for Deeper Mathematics" by Matthias Beck and Ross Geoghegan

We hope that this article has been helpful in answering some common questions about the problem of proving that the expression $3^{2n} - 4$ is a multiple of 5 for all natural numbers $n$.