Problem 42 - Discrete Mathematics with Applications

Question

Let \(a, b, c,\) and \(n\) be any positive integers and \(p\) be any prime. Prove each. Let \(a | c\) and \(b | c,\) where \(a\) and \(b\) are relatively prime numbers. Then \(a b | c\).

Step-by-Step Solution

Answer

Since \(a\) and \(b\) are relatively prime, their greatest common divisor is 1. We can express their gcd as a linear combination, \(gcd(a, b) = ax + by = 1\), where \(x\) and \(y\) are integers. Given that \(a|c\) and \(b|c\), we have integers \(m\) and \(n\) such that \(c = am\) and \(c = bn\). We can express \(c\) as a linear combination of \(am\) and \(bn\): \(c = axm + byn\). Factoring out \(ab\) gives \(c = ab(\frac{x}{b}m + \frac{y}{a}n)\). Since \(\frac{x}{b}m + \frac{y}{a}n\) is an integer, we can conclude that \(ab | c\). Therefore, if \(a|c\) and \(b|c\), where \(a\) and \(b\) are relatively prime numbers, then \(ab | c\).

Step 1: Find gcd and Linear Combination of a and b

Since \(a\) and \(b\) are relatively prime, their greatest common divisor is 1. Therefore, by the Extended Euclidean Algorithm, we can express their gcd as a linear combination of \(a\) and \(b\): \[gcd(a, b) = ax + by = 1\] where \(x\) and \(y\) are integers.

Step 2: Determine Divisibility of a and b

Given that \(a | c\) and \(b | c\), we know that there exist integers \(m\) and \(n\) such that: \[c = am\] \[c = bn\]

Step 3: Express c as a Linear Combination of am and bn

Using the two equalities from step 2, we can express \(c\) as a linear combination of \(am\) and \(bn\): \[c = axm + byn\]

Step 4: Factor out ab from the Resulting Linear Combination

Now, we factor out \(ab\) from the linear combination in step 3: \[c = a(xm) + b(yn) = ab(\frac{x}{b}m + \frac{y}{a}n)\]

Step 5: Prove ab | c

From step 4, we can see that \(c\) is a multiple of \(ab\), as it is the product of \(ab\) and the term in parentheses. Since \(\frac{x}{b}m + \frac{y}{a}n\) is an integer (because \(m\), \(n\), \(x\), \(y\) are integers and \(a|c\) and \(b|c\)), we can conclude that \(ab | c\). Therefore, if \(a | c\) and \(b | c\), where \(a\) and \(b\) are relatively prime numbers, then \(ab | c\).

Key Concepts

Prime Numbers

Prime numbers are the backbone of many concepts in number theory, and they are integers greater than 1 that have no divisors other than 1 and themselves. In other words, a prime number can only be divided evenly (without a remainder) by 1 and the number itself.

Understanding the role of prime numbers is crucial when dealing with divisibility and the factorization of integers. For example, if two numbers are relatively prime, like in our exercise scenario, it means that the only positive integer that divides both of them is 1. This unique property leads us towards understanding why the product of two relatively prime numbers divides another number if they individually do so.

Greatest Common Divisor

The greatest common divisor (GCD) of two numbers is the largest positive integer that divides both of them without leaving a remainder. It plays a significant role when determining the divisibility between numbers.

For any two integers, if their GCD is 1, they are said to be coprime or relatively prime, as seen in our exercise. The fact that relatively prime numbers have a GCD of 1 means that they share no prime factors, and it simplifies the examination of their multiples' relationships, particularly in divisibility problems.

Extended Euclidean Algorithm

The Extended Euclidean Algorithm is a powerful tool for finding not only the greatest common divisor of two integers but also the coefficients that express the GCD as a linear combination of these integers. This is represented by the equation \[gcd(a, b) = ax + by\]where in our problem, since the GCD is 1, the equation simplifies to \[1 = ax + by\].

The algorithm extends the basic Euclidean Algorithm by backtracking the steps used to find the GCD to provide us with the particular integers x and y. This method is invaluable when solving Diophantine equations or proving properties such as in the given divisibility problem.

Linear Combination

A linear combination in mathematics is an expression constructed from a set of terms by multiplying each term by a constant and adding the results. In the context of integer divisibility, a linear combination helps us understand how a new number can be formed from the multiples of existing numbers we are analyzing.

In the original exercise, we used the concept of a linear combination to express the integer c as a multiple of both a and b. By doing so, we demonstrated that \[c = a(xm) + b(yn) = ab\left(\frac{x}{b}m + \frac{y}{a}n\right)\],proving the relationship between the divisibility by a and b when they are relatively prime and consequently the divisibility by their product ab. This interplay between linear combinations and divisibility is a fundamental concept that provides insight into solving various problems related to number theory.