Problem 7 - A Book of Abstract Algebra

Question

If \(\operatorname{lcm}(a, b)=a b\), then \(\operatorname{gcd}(a, b)=1\)

Step-by-Step Solution

Answer

Given that \(\operatorname{lcm}(a, b) = a \cdot b\), it follows that \(\operatorname{gcd}(a, b) = 1\).

Step 1: Understanding LCM

The least common multiple (LCM) of two numbers, \(a\) and \(b\), is the smallest positive integer that is divisible by both \(a\) and \(b\). When \(\operatorname{lcm}(a, b) = a \cdot b\), this suggests that \(a\) and \(b\) do not share any common prime factors. Consequently, their least common multiple is simply their product.

Step 2: Understanding GCD

The greatest common divisor (GCD) of two numbers, \(a\) and \(b\), is the largest positive integer that divides both \(a\) and \(b\). When two numbers have no common factors other than 1, their GCD is 1, indicating they are coprime.

Step 3: Linking LCM and GCD

The relationship between the LCM and GCD of two numbers can be expressed by the formula: \[ \operatorname{lcm}(a, b) \times \operatorname{gcd}(a, b) = a \times b \]. Given that \(\operatorname{lcm}(a, b) = a \times b\), substituting in the formula provides \[ (a \times b) \times \operatorname{gcd}(a, b) = a \times b \].

Step 4: Solving for GCD

Since \( (a \times b) \times \operatorname{gcd}(a, b) = a \times b \), dividing both sides by \(a \times b\) shows that \(\operatorname{gcd}(a, b) = 1\). Thus, \(a\) and \(b\) are coprime.

Key Concepts

Coprime Numbers

Coprime numbers are two or more integers that have no common factors other than 1. This means that the greatest common divisor (GCD) of coprime numbers is always 1. For example, take the numbers 8 and 15. The factors of 8 are 1, 2, 4, and 8. The factors of 15 are 1, 3, 5, and 15. The only common factor between these two sets is 1, making 8 and 15 coprime numbers.
The concept of coprime numbers is crucial in understanding the relationship between the least common multiple (LCM) and GCD. When the LCM of two numbers, like in the exercise above, equals their product, these numbers have no common factors other than 1. This is why they are defined as coprime. Coprime numbers are also sometimes referred to as relatively prime numbers.
In mathematical problems, recognizing coprime numbers can help quickly determine solutions, especially when dealing with LCM and GCD. It's a simple yet powerful concept used frequently in number theory and problem-solving.

Least Common Multiple

The least common multiple (LCM) of two integers is the smallest number that can be evenly divided by both. Imagine two gears with different numbers of teeth rotating together. The LCM is like finding the first time they align perfectly.

• For example, consider the numbers 4 and 6. The multiples of 4 are 4, 8, 12, 16, and so on. The multiples of 6 are 6, 12, 18, and so on. The smallest multiple they share is 12, so the LCM of 4 and 6 is 12.

When talking about coprime numbers, their LCM will simply be the product of the numbers themselves, like in our original exercise where \( \operatorname{lcm}(a, b) = a \cdot b \). Because these numbers are coprime, they don't share factors that would otherwise decrease the LCM.
The LCM is a key concept in simplifying fractions and solving problems involving periodic events or synchronization in operations.

Greatest Common Divisor

The greatest common divisor (GCD), sometimes called greatest common factor, is the largest positive integer that divides each of the numbers in question without leaving a remainder. Think of the GCD as the maximum number of identical groups you can divide two amounts into without any leftovers. To find the GCD, you list the factors of each number and take the greatest one they have in common.

• For example, with the numbers 28 and 35, the factors of 28 are 1, 2, 4, 7, 14, and 28, while the factors of 35 are 1, 5, 7, and 35. Both lists share the number 7 as the largest common factor, making it their GCD.

In the context of the exercise, when two numbers are coprime, their GCD is 1. This means there are no prime factors they share. The formula \[ \operatorname{lcm}(a, b) \times \operatorname{gcd}(a, b) = a \times b \] helps illustrate why the GCD is 1 when the LCM equals the product of coprime numbers. Recognizing the GCD is essential in simplifying algebraic expressions and solving equations in mathematics.