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 has a unique prime factorization with even exponents.

Step 1: Understand the Problem

We need to show that the square of any natural number greater than 1 can be expressed as a product of primes, where each prime appears an even number of times.

Step 2: Recall the Fundamental Theorem of Arithmetic

The Fundamental Theorem of Arithmetic states that every natural number greater than 1 can be uniquely represented as a product of prime numbers, up to the order of the factors. This implies that each natural number has a unique prime factorization.

Step 3: Prime Factorization of a Number

Take a natural number greater than 1, say \( n \). According to the theorem, \( n \) can be written uniquely as a product of prime factors: \( n = p_1^{a_1} \times p_2^{a_2} \times \cdots \times p_k^{a_k} \) where \( p_1, p_2, \ldots, p_k \) are distinct primes and \( a_1, a_2, \ldots, a_k \) are positive integers.

Step 4: Square the Number

Now, calculate \( n^2 = (p_1^{a_1} \times p_2^{a_2} \times \cdots \times p_k^{a_k})^2 \). Applying the rules of exponents, this expression becomes \( p_1^{2a_1} \times p_2^{2a_2} \times \cdots \times p_k^{2a_k} \).

Step 5: Verify Even Exponents

In the factorization of \( n^2 \), each prime \( p_i \) is raised to the power \( 2a_i \). Since \( 2a_i \) is clearly an even number (as it is 2 times any integer \( a_i \)), every prime number is indeed occurring an even number of times in the product.

Step 6: Conclude Uniqueness and Evenness

Since the original factorization of \( n \) was unique, \( n^2 \) inherits this uniqueness in its factorization, except for the order of primes, with all exponents being even, thus satisfying the conditions of the problem.

Key Concepts

Prime Factorization

Prime factorization is the process of breaking down a natural number into a product of prime numbers. A prime number is a natural number greater than 1 that is not divisible by any other natural numbers except for 1 and itself. For example, the number 18 can be factorized into primes as 2 and 3, since 18 = 2 × 3 × 3.
Understanding prime factorization is essential because it is the bedrock of the Fundamental Theorem of Arithmetic. This theorem assures us that every natural number greater than 1 has a unique prime factorization.

• Example: 45 = 3 × 3 × 5
• Prime numbers only: cannot be further divided
• Foundation for more complex calculations

Leveraging prime factorization helps us solve various mathematical problems, including checking divisibility, finding the greatest common divisor (GCD), and least common multiple (LCM).

Natural Numbers

Natural numbers are a fundamental concept in mathematics, representing the set of all positive integers starting from 1. They are typically used for counting and ordering and are an infinite set.
Natural numbers include numbers like 1, 2, 3, 4, and so on, but do not include zero or negative numbers.

• Begin from 1 and continue infinitely
• Used for basic counting and operations
• Key component in arithmetic operations

Natural numbers often play a central role in various mathematical theories and concepts, including the Fundamental Theorem of Arithmetic. This theorem is applicable only to natural numbers greater than 1 when discussing prime factorization and its uniqueness.

Exponents

Exponents represent a shorthand way of expressing repeated multiplication of a number by itself. For example, writing 3 multiplied by itself 4 times as 3⁴. This notation helps simplify large expressions and makes calculations easier.
In prime factorization, exponents are used to indicate how many times a prime number appears as a factor of the original number. For example, in the factorization 81 = 3 × 3 × 3 × 3, we write it as 3⁴, emphasizing the repeated prime factor.

• Simplifies repeated multiplication
• Important in prime factorization for clarity
• Expresses large products easily

Understanding exponents is crucial in the Fundamental Theorem of Arithmetic, especially when exploring the squares of numbers, as each exponent is doubled, ensuring all primes in the factorization of the square have even exponents.

Unique Factorization

Unique factorization, as guaranteed by the Fundamental Theorem of Arithmetic, means that each natural number greater than 1 can be represented in exactly one way as a product of prime numbers, aside from the order of these factors. This uniqueness is a cornerstone of number theory.
When a number is factorized into primes, no two distinct sets of primes can give the same number through multiplication. This property is especially important when considering the square of a natural number.

• One prime factorization per natural number
• Order doesn’t affect uniqueness
• Critical for mathematical proofs and theories

In practical terms, understanding unique factorization enables mathematicians to efficiently solve problems related to divisibility, congruences, and more within number theory, relying on the certitude that our factorizations are not ambiguous.