Problem 21 - Algebra: Pure and Applied

Question

Find a generator of the subgroup \(6 \mathbb{Z} \cap 15 \mathbb{Z}\) of \(\mathbb{Z}\).

Step-by-Step Solution

Answer

The generator of the subgroup is 30.

Step 1: Understand What a Generator Is

A generator of a subgroup in the integers is a number that allows all elements in the subgroup to be expressed as multiples of it. Essentially, it is the smallest positive integer in the subgroup that generates all other elements.

Step 2: Define the Subgroup

The subgroup in question is defined by the intersection of two subgroups: \(6 \mathbb{Z}\) and \(15 \mathbb{Z}\). This means we are looking for integers that are multiples of both 6 and 15.

Step 3: Find the Least Common Multiple (LCM)

The generator of the intersection can be found by calculating the greatest common divisor (GCD) of the given numbers (6 and 15). The number that generates the intersection is precisely the LCM, not the GCD. First, calculate the prime factorizations. 6 is \(2 \times 3\) and 15 is \(3 \times 5\).

Step 4: Calculate the Least Common Multiple

To find the LCM, for each prime number, take the highest power present in either factorization. The LCM of \(6\) and \(15\) is \(2^1 \times 3^1 \times 5^1 = 30\).

Step 5: Verify the Generator

Verify that 30 is a multiple of both 6 and 15 by dividing it by each number. \(30 \div 6 = 5\) and \(30 \div 15 = 2\), both results are integers. Therefore, 30 is indeed a common multiple and the smallest integer that can generate the subgroup.

Key Concepts

Least Common Multiple

The Least Common Multiple (LCM) of two numbers is the smallest positive integer that is divisible by both of them. This is a crucial concept when solving problems involving arithmetic on subgroups of integers. To find the LCM of two numbers:

• Perform the prime factorization of each number.
• Identify all the prime numbers involved.
• For each prime number, choose the highest power that appears in either of the factorizations.
• Multiply these highest powers together to get the LCM.

Understanding LCM helps in determining a generator for the intersection of two subgroups. For instance, to find a generator of the subgroup obtained by intersecting the multiples of 6 and 15 (\(6 \mathbb{Z} \cap 15 \mathbb{Z}\)), you calculate the LCM of 6 and 15. The calculated LCM, 30, is the smallest integer that is a common multiple of both numbers. Therefore, 30 serves as the generator of the intersection of these subgroups.

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's a key tool in understanding divisibility and factorization problems. Here's how you can find the GCD:

• List the prime factors of each number.
• Identify the common prime factors shared by both lists.
• Take the lowest power for each shared prime factor.
• Multiply these lowest powers together to get the GCD.

While in some contexts, like finding common factors or reducing fractions, the GCD is pivotal, determining a generator of the intersection of subgroups like \(6 \mathbb{Z} \cap 15 \mathbb{Z}\) requires using the LCM instead. The GCD helps us find these common factors, but the generator for the intersection ultimately comes from the LCM of the given numbers.

Intersection of Subgroups

When we talk about the intersection of subgroups, we're referring to the set of elements that are common to both subgroups. In the context of integer subgroups like \(6 \mathbb{Z}\) and \(15 \mathbb{Z}\), the intersection represents integers that are multiples of both 6 and 15. To express this intersection concretely:

• Determine the generator of the intersection by calculating the LCM of the integers defining the subgroups.
• The intersection consists of all integer multiples of this LCM.

By finding the LCM, we establish a clean and structured method to define the common elements of the intersecting subgroups. Hence, in this example, our focus on the LCM helps identify 30 as the generator of \(6 \mathbb{Z} \cap 15 \mathbb{Z}\), leading to a clear understanding of which integers fall into this intersection. This approach allows you to easily calculate and comprehend similar subgroup intersections with other pairs of integers.