Problem 76 - Calculus Early Transcendentals: Pearson New International Edition

Question

. Use the Fundamental Theorem of Arithmetic (Problem 75) to show that the square of any natural number greater than 1 can be written as the product of primes in a unique way, except for the order of the factors, with each prime occurring an even number of times. For example, \((45)^{2}=3 \cdot 3 \cdot 3 \cdot 3 \cdot 5 \cdot 5\).

Step-by-Step Solution

Answer

The square of any natural number greater than 1 can be expressed uniquely as a product of primes, each appearing an even number of times.

Step 1: Understanding the Fundamental Theorem of Arithmetic

The Fundamental Theorem of Arithmetic states that every natural number greater than 1 can be represented uniquely as a product of prime numbers. For example, 45 can be expressed as the product of primes: \(45 = 3 \times 3 \times 5\).

Step 2: Express the Natural Number as Product of Primes

Given any natural number \(n > 1\), it can be decomposed into its prime factors. Let \(n = p_1^{a_1} \times p_2^{a_2} \times \ldots \times p_k^{a_k}\), where \(p_i\) are prime numbers and \(a_i\) are their respective powers. This decomposition is unique up to the order of the factors.

Step 3: Square the Prime Factorization

To find \(n^2\), we square each prime in the factorization of \(n\): \[(n^2) = (p_1^{a_1})^2 \times (p_2^{a_2})^2 \times \ldots \times (p_k^{a_k})^2 = p_1^{2a_1} \times p_2^{2a_2} \times \ldots \times p_k^{2a_k}.\]

Step 4: Analyzing Prime Factor Powers in the Square

In the resulting factorization of \(n^2\), each prime \(p_i\) appears \(2a_i\) times. Since \(2a_i\) is always even for any integer \(a_i\), the prime factorization of \(n^2\) will have each prime occurring an even number of times.

Step 5: Conclude the Uniqueness of the Prime Factorization

Since each prime \(p_i\) appears exactly \(2a_i\) times, and the factorization is unique up to the order of the primes, the product of prime numbers obtained for \(n^2\) is unique. Thus, every square of a natural number has its prime factorization with primes occurring an even number of times.

Key Concepts

Prime Factorization

When we talk about the prime factorization of a number, we refer to the process of breaking down a number into its basic building blocks. In mathematics, these basic building blocks are prime numbers. A prime number is a whole number greater than 1 that cannot be exactly divided by any whole number other than itself and 1. For example, the number 12 can be factored into the primes 2 and 3 as it equals \(2 \times 2 \times 3\). Prime factorization is quite powerful because every natural number greater than 1 is either a prime itself or can be decomposed into prime factors. This idea is the backbone of the Fundamental Theorem of Arithmetic.

• Start with the smallest prime that divides the number.
• Divide the number by this prime and continue the process with the quotient.
• Repeat until the only remaining factor is a prime number.

This unique breakdown is consistent and provides the same set of prime factors regardless of the order you find them in.

Natural Numbers

Natural numbers are the simplest and most basic numbers in the number system. They're the numbers we typically learn as children when counting objects: 1, 2, 3, and so on. Notably, natural numbers do not include negative numbers or fractions; they are always whole and positive. These numbers serve as the basis for more complex mathematical concepts, including prime factorization.When working with natural numbers greater than 1, each can be broken into a product of prime numbers. For example, number 28 can be broken into \(2 \times 2 \times 7\).This decomposition is crucial for understanding the properties of the numbers themselves, including why the square of a natural number has a unique prime factorization.

Unique Product of Primes

The expression of natural numbers as a unique product of primes is a fundamental idea in mathematics. This means that for any natural number greater than 1, there is only one set of prime factors (considering their multiplicities) that builds the number. For instance, 60 can be exclusively expressed as \(2^2 \times 3 \times 5\).With the Fundamental Theorem of Arithmetic, we know this product's uniqueness ensures consistency. This theorem posits that the prime factorization of any number is unique, except for the order of the factors.When we square a number, we take each prime factor, raise it to twice its original power, and multiply them together. This uniqueness helps us understand how different numbers relate and ensures that complex calculations are reliable throughout mathematics and applications.

Square of a Number

When you square a number, you multiply it by itself. For instance, the square of 5 is \(5 \times 5 = 25\). An interesting property emerges when we look at the prime factorization of the square of a number. To get the square of a natural number, simply take each prime in its factorization and double its power. So if \(n=p_1^{a_1} \times p_2^{a_2}\), then \(n^2=p_1^{2a_1} \times p_2^{2a_2}\). This process results in each prime number factor appearing an even number of times. Here's why a square's prime factorization is special:

• Every prime occurs with an even exponent.
• This evenness ensures there's no ambiguity - different numbers produce different squares.

Understanding this allows us to appreciate how prime factors behave under multiplication, confirming that squaring doesn't change the number's fundamental nature but instead builds on it predictably.