Problem 6 - A Book of Abstract Algebra

Question

Let \(\operatorname{gcd}(a, b)=c .\) Write \(a=c a^{\prime}\) and \(b=c b^{\prime} .\) Then $$ \operatorname{gcd}\left(a^{\prime}, b^{\prime}\right)=\operatorname{gcd}\left(a, b^{\prime}\right)=\operatorname{gcd}\left(a^{\prime}, b\right)=1 $$

Step-by-Step Solution

Answer

All expressions simplify by proving shared divisibility conflicts with the maximal gcd assumption, confirming 1 is the greatest common divisor for each.

Step 1: Understanding the Exercise

We start with the given situation: the greatest common divisor (gcd) of two numbers \(a\) and \(b\) is \(c\). Thus, both \(a\) and \(b\) can be rewritten in terms of \(c\) and new variables \(a'\) and \(b'\), such that \(a = c \cdot a'\) and \(b = c \cdot b'\). The goal is to show that certain gcd equations involving \(a'\) and \(b'\) equal 1.

Step 2: Expressing in Terms of New Variables

Since \(a = c \cdot a'\) and \(b = c \cdot b'\), \(c\) divides both \(a\) and \(b\). The fact that \(c\) is the gcd of \(a\) and \(b\) means \(c\) is the largest number dividing both. Consequently, any factor common to \(a'\) and \(b'\) must not affect \(a\) or \(b\) beyond \(c\), because any common factor of \(a'\) and \(b'\) would contravene the gcd condition of \(a\) and \(b\).

Step 3: Proving \(\operatorname{gcd}(a', b') = 1\)

Since \(a = c a'\) and \(b = c b'\), any common factor of \(a'\) and \(b'\) would make \(c\) not maximal if it were greater than 1, contradicting \(c = \operatorname{gcd}(a, b)\). Thus, the only possible common divisor of \(a'\) and \(b'\) is 1: \(\operatorname{gcd}(a', b') = 1\).

Step 4: Proving \(\operatorname{gcd}(a, b') = 1\) and \(\operatorname{gcd}(a', b) = 1\)

To show \(\operatorname{gcd}(a, b') = 1\), assume a common divisor \(d\) exists for \(a = c a'\) and \(b'\). If so, then \(d\) divides \(c a'\) and thus \(c\) and \(a'\). Since \(c\) already divides both \(a\) and \(b\), and \(b'\) is free of additional factor beyond \(c\), it follows \(d\) must be 1.Similarly, for \(\operatorname{gcd}(a', b) = 1\), a common divisor \(d\) of \(a'\) and \(b = c b'\) implies \(d\) divides \(ca'\) and \(b\), restricting \(d\) to involvement only through \(c\). Hence, \(d\) must be 1.

Key Concepts

Integer Factorization

Integer factorization involves breaking down a composite number into a product of smaller integers or prime factors. When given two numbers, say \(a\) and \(b\), if their greatest common divisor (GCD) is \(c\), then we can express these numbers like this:

• \(a = c \cdot a'\)
• \(b = c \cdot b'\)

Here, \(a'\) and \(b'\) serve as the factors of \(a\) and \(b\) when they've been divided by their GCD, \(c\).
This breakdown is useful because it helps isolate the shared component of \(a\) and \(b\) (which is \(c\)), allowing us to consider what's left (\(a'\) and \(b'\)) independently. By understanding this process, it becomes easier to work with these numbers and apply further number theory analyses, such as determining coprime relationships.

Coprime Integers

Coprime integers, or relatively prime integers, are pairs of numbers whose greatest common divisor is 1. In this exercise, the pair \((a', b')\) are elements that arise from integer factorization.

• Since \(a = c \cdot a'\) and \(b = c \cdot b'\) with \(c\) as \(\operatorname{gcd}(a, b)\), any common factor between \(a'\) and \(b'\) would indicate a higher GCD than \(c\), which cannot be true.
• Therefore, \(\operatorname{gcd}(a', b') = 1\).

Understanding that \(a'\) and \(b'\) are coprime is crucial: it emphasizes the uniqueness of \(c\) as the sole common factor for \(a\) and \(b\). It also means that any attempt to find a common divisor for \(a'\) and \(b'\) would result in 1, further backing their coprimeness. This is important in number theory because coprime values simplify many operations and calculations, like finding least common multiples.

Number Theory Concepts

Number theory is an area of mathematics dedicated to studying integers and their properties. Number theory concepts such as the GCD, integer factorization, and coprimeness offer insight into the structure and relations of numbers. Deep understanding comes with the application of:

• GCD: Utilizing the greatest common divisor to simplify and solve complex integer problems.


• Factorization: Decomposing integers to manage and analyze complex expressions.


• Coprimeness: Playing a crucial role in developing further mathematical properties, easing calculations with relatively prime numbers.

Each of these concepts connects to others and builds an interconnected framework that enables solving problems involving divisibility and primes. For example, proving that certain relationships among numbers or their components hold reinforces the notion that integer properties can become axioms for further mathematical theory. Practical applications in encryption, coding theory, and algebra benefit significantly from solid grounding in these number theory concepts.