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Chapter 8: Problem 3

Use mathematical induction to prove the second part of Corollary \(8.14 .\) Let \(p\) be a prime number, let \(n \in \mathbb{N},\) and let \(a_{1}, a_{2}, \ldots, a_{n} \in \mathbb{Z}\). If \(p \mid\left(a_{1} a_{2} \cdots a_{n}\right),\) then there exists a \(k \in \mathbb{N}\) with \(1 \leq k \leq n\) such that \(p \mid a_{k}\)

Short Answer

Expert verified
We will use mathematical induction to prove that if \(p\) is a prime number and \(p\) divides the product of a sequence of \(n\) integers \((a_1, a_2, ..., a_n)\), there must be at least one integer in the sequence such that \(p\) divides the integer. Base case: For \(n = 1\), we have \(p \mid a_1\), so the statement holds. Inductive hypothesis: Assume the statement is true for \(n = k\). Inductive step: We want to prove it for \(n = k + 1\). We have \(p \mid (a_1 a_2 \cdots a_{k} a_{k+1})\). There are two possibilities: (1) \(p \mid a_{k+1}\), in which case the statement holds; (2) \(p \nmid a_{k+1}\), in which case \(p\) must divide the other factor in the product, i.e., \(p \mid (a_1 a_2 \cdots a_{k})\). By our inductive hypothesis, there exists an integer \(1 \leq m \leq k\) such that \(p \mid a_m\), so the statement holds for \(n = k + 1\). Thus, by mathematical induction, we have proved the statement for any \(n\in\mathbb{N}\).

Step by step solution

01

Base case: n = 1

In this case, we only have one integer \(a_1\), and we're given that \(p \mid a_1\). Since \(p\) already divides \(a_1\), we have proved the base case.
02

Inductive hypothesis

Now, let us assume the statement is true for some \(n = k\), i.e., if \(p \mid (a_1 a_2 \cdots a_k)\), then there exists an integer \(1 \leq m \leq k\) such that \(p \mid a_m\). We want to show that this holds for \(n = k + 1\), i.e., if \(p \mid (a_1 a_2 \cdots a_{k} a_{k+1})\), then there exists an integer \(1 \leq m \leq k + 1\) such that \(p \mid a_m\).
03

Inductive step

We have \(p \mid (a_1 a_2 \cdots a_{k} a_{k+1})\), so we can rewrite the left-hand side as: \(p \mid a_{k+1}(a_1 a_2 \cdots a_{k})\) Now, we have two possibilities: 1. \(p \mid a_{k+1}\) 2. \(p \nmid a_{k+1}\) In case (1), we have already found an integer in the sequence such that \(p \mid a_{k+1}\) and in this case our statement holds for \(n=k+1\). In case (2), since \(p\nmid a_{k+1}\) and \(p\) is a prime, p must divide the other factor in the product i.e, \(p \mid (a_1 a_2 \cdots a_{k})\). By our inductive hypothesis, since \(p \mid (a_1 a_2 \cdots a_{k})\), we know that there exists an integer \(1 \leq m \leq k\) such that \(p \mid a_m\). This means our statement holds for \(n = k + 1\) as well. Therefore, by mathematical induction, we have proved that if \(p\) is a prime number and \(p\) divides the product of a sequence of \(n\) integers, there must be at least one integer in the sequence such that \(p\) divides the integer.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Prime Numbers
Prime numbers are fascinating and fundamental in mathematics. A prime number is defined as a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. In other words, a prime number can only be divided evenly by 1 and itself.

Here are some key points to remember about prime numbers:
  • The smallest prime number is 2, which is also the only even prime number. Every other even number can be divided by 2, so they are not prime.
  • Prime numbers play a crucial role in number theory due to their divisibility properties.
  • They are the building blocks of numbers, much like atoms are for molecules in chemistry.
  • There are infinitely many primes, as proven by Euclid in ancient times.
Understanding prime numbers helps not only in solving mathematical problems but also in fields like cryptography, where primes are used in keys to secure data.
Divisibility
Divisibility is the concept of one number being able to divide another exactly, leaving no remainder. This concept is essential when working with sequences and products in mathematics.

Some important facts about divisibility:
  • A number, say \(a\), is divisible by another number \(b\), if there exists an integer \(c\) such that \(a = bc\).
  • If \(p\) is a prime and it divides a product of integers, \(p\) must divide at least one of those integers. This is known as a key property of primes.
  • Divisibility can simplify complex problems, especially in proving mathematical statements, such as those involving integer sequences or algebraic expressions.
Understanding divisibility improves problem-solving abilities and helps in recognizing patterns within numbers and sequences.
Natural Numbers
Natural numbers are the set of positive integers beginning from 1, 2, 3, and so on. They are the simplest set of numbers used in mathematics and form the foundation for most number systems. Here's what you need to know:

  • Natural numbers can represent quantities like objects, ranking, and counting. They serve as the stepping stones to understanding more complex number systems.
  • This set does not include zero or negative numbers but can be extended to whole numbers, which do.
  • Operations like addition and multiplication are always closed within natural numbers, meaning the result is also a natural number.
  • Mathematical induction often relies on natural numbers, especially for proving statements about sequences or sets.
Natural numbers are integral to all areas of mathematics, serving as a basis for introductory math concepts to advanced theorems.
Integer Sequences
An integer sequence is a list of numbers where each number is an integer. These sequences can be finite or infinite and often follow a specific pattern or rule. Let鈥檚 explore some crucial aspects:

  • Integer sequences can include both positive and negative integers, enabling more flexibility and applications in various mathematical problems.
  • Sequences are used to identify patterns, prove theorems, and solve equations, often appearing in different areas like algebra and calculus.
  • Prime numbers can be found within integer sequences, such as the sequence of prime numbers or a defined sequence where each number is a result of a specific formula or condition.
  • Understanding sequences involves grasping concepts like arithmetic and geometric progressions, recursive relations, and even divisibility rules.
Working with integer sequences fosters a deeper comprehension of numbers and their relationships, making sequential reasoning a vital skill in mathematics.

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Most popular questions from this chapter

Is the following proposition true or false? Justify your conclusion. $$ \text { If } n \in \mathbb{N}, \text { then } \operatorname{gcd}(5 n+2,12 n+5)=1 \text { . } $$

Square Roots and Irrational Numbers. In Chapter \(3,\) we proved that some square roots (such as \(\sqrt{2}\) and \(\sqrt{3}\) ) are irrational numbers. In this activity, we will use the Fundamental Theorem of Arithmetic to prove that if a natural number is not a perfect square, then its square root is an irrational number. (a) Let \(n\) be a natural number. Use the Fundamental Theorem of Arithmetic to explain why if \(n\) is composite, then there exist distinct prime numbers \(p_{1}, p_{2}, \ldots, p_{r}\) and natural numbers \(\alpha_{1}, \alpha_{2}, \ldots, \alpha_{r}\) such that $$ n=p_{1}^{\alpha_{1}} p_{2}^{\alpha_{2}} \cdots p_{r}^{\alpha_{r}} $$ Using \(r=1\) and \(\alpha_{1}=1\) for a prime number, explain why we can write any natural number greater than one in the form given in equation (1). (b) A natural number \(b\) is a perfect square if and only if there exists a natural number \(a\) such that \(b=a^{2}\). Explain why \(36,400,\) and 15876 are perfect squares. Then determine the prime factorization of these perfect squares. What do you notice about these prime factorizations? (c) Let \(n\) be a natural number written in the form given in equation (1) in part (a). Prove that \(n\) is a perfect square if and only if for each natural number \(k\) with \(1 \leq k \leq r, \alpha_{k}\) is even. (d) Prove that for all natural numbers \(n,\) if \(n\) is not a perfect square, then \(\sqrt{n}\) is an irrational number. Hint: Use a proof by contradiction.

Let \(a, b,\) and \(c\) be integers with \(a \neq 0\) and \(b \neq 0 .\) If \(a\) and \(b\) are relatively prime, then the linear Diophantine equation \(a x+b y=c\) has infinitely many solutions. In addition, if \(\left(x_{0}, y_{0}\right)\) is a particular solution of this equation, then all the solutions of the equation are given by $$x=x_{0}+b k \quad y=y_{0}-a k$$ where \(k \in \mathbb{Z}\).

The goal of this exercise is to determine all (integer) solutions of the linear Diophantine equation in three variables \(12 x_{1}+9 x_{2}+16 x_{3}=20\) (a) First, notice that gcd \((12,9)=3\). Determine formulas that will generate all solutions for the linear Diophantine equation \(3 y+16 x_{3}=20\) (b) Explain why the solutions (for \(x_{1}\) and \(x_{2}\) ) of the Diophantine equation \(12 x_{1}+9 x_{2}=3 y\) can be used to generate solutions for \(12 x_{1}+9 x_{2}+16 x_{3}=20\) (c) Use the general value for \(y\) from Exercise (6a) to determine the solutions of \(12 x_{1}+9 x_{2}=3 y\) (d) Use the results from Exercises (6a) and (6c) to determine formulas that will generate all solutions for the Diophantine equation \(12 x_{1}+9 x_{2}+16 x_{3}=20\) Note: These formulas will involve two arbitrary integer parameters. Substitute specific values for these integers and then check the resulting solution in the original equation. Repeat this at least three times. (e) Check the general solution for \(12 x_{1}+9 x_{2}+16 x_{3}=20\) from Exercise (6d).

The Twin Prime Conjecture states that there are infinitely many twin primes, but it is not known if this conjecture is true or false. The answers to the following questions, however, can be determined. (a) How many pairs of primes \(p\) and \(q\) exist where \(q-p=3\) ? That is, how many pairs of primes are there that differ by 3 ? Prove that your answer is correct. (One such pair is 2 and \(5 .\) ) (b) How many triplets of primes of the form \(p, p+2,\) and \(p+4\) are there? That is, how many triplets of primes exist where each prime is 2 more than the preceding prime? Prove that your answer is correct. Notice that one such triplet is \(3,5,\) and 7
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