Problem 5 - A Book of Abstract Algebra

Question

\(\operatorname{lcm}(a, a b)=a b\)

Step-by-Step Solution

Answer

The least common multiple of \(a\) and \(ab\) is \(ab\).

Step 1: Understand the Expression

We are given the expression \(\operatorname{lcm}(a, ab) = ab\). This suggests the least common multiple of \(a\) and \(ab\) is equal to \(ab\).

Step 2: Explore LCM Definition

The least common multiple (LCM) of two numbers is the smallest number divisible by both. For two numbers \(x\) and \(y\), if \(y\) is a multiple of \(x\), then \(\operatorname{lcm}(x, y) = y\).

Step 3: Apply LCM Property

Since \(ab\) is a multiple of \(a\), we have \(\operatorname{lcm}(a, ab) = ab\). This agrees with the LCM definition when one number is a multiple of the other.

Step 4: Validate the Result

Check if \(ab\) is indeed the LCM. Since any multiple of \(a\) and \(ab\) must include the factors that compose \(ab\), \(ab\) itself is the smallest multiple containing both \(a\) and \(ab\). Thus, \(a\) divides \(ab\), and \(ab\) is the LCM.

Key Concepts

Multiples

Before diving into the concept of the least common multiple, it's essential to understand what multiples are. A multiple of a number is simply what you get when you multiply it by an integer. For example, multiples of 3 are 3, 6, 9, and so on. This pattern continues indefinitely.

Multiples are the building blocks for understanding LCM. When looking for the LCM of two numbers, you try to find the smallest multiple they have in common. In our problem, we're looking at numbers like \(a\) and \(ab\). Since \(ab\) includes \(a\) as a factor multiplied by another factor \(b\), \(ab\) is naturally a multiple of \(a\). This idea is crucial as it simplifies the calculation of LCM in such scenarios.

Divisibility

Divisibility is what determines if one number can be divided by another without leaving a remainder. In simple terms, if a number \(x\) can be exactly divided by \(y\), then \(y\) is a divisor of \(x\).

For example, 10 is divisible by 2 because it can be divided evenly, leaving no remainder. When considering the LCM, we always look for the smallest number \(z\) that can be divided by both numbers in question without remainder. In our exercise, \(ab\) being the LCM implies it is the smallest number divisible by both \(a\) and \(ab\).

• Why \(ab\)? Because in \(ab\), every factor required to divide both \(a\) and \(ab\) is present in minimal form.
• \(a\) divides \(ab\) naturally since \(ab = a\times b\), and \(b\) is just a factor multiplying \(a\).

Mathematical Problem Solving

Mathematical problem solving often involves breaking down a problem into more manageable parts. The exercise of determining the least common multiple (LCM) uses a strategic approach that connects understanding of multiples and divisibility.

To solve LCM problems effectively, you may need to follow steps:

• Identify Multiples: Determine the multiples of the given numbers.
• Check Divisibility: Look for numbers that both original numbers can divide evenly.
• Find the Smallest Common Multiple: Among the shared multiples, identify the smallest one.

In our precise example of \( lcm(a, ab) = ab \), recognizing that \(ab\) as a multiple of \(a\) is key. It allows us to apply the rule that when one number is a multiple of another, that larger number is inherently the LCM, simplifying the problem greatly.

By understanding these concepts, students can approach mathematical problems with confidence, knowing when and how to apply principles of multiples and divisibility effectively in problem-solving scenarios.