Problem 10 - Fundamentals of Mathematics

Question

Find the first five common multiples of the following numbers. 4 and 5

Step-by-Step Solution

Answer

The first five common multiples of 4 and 5 are 20, 40, 60, 80, and 100.

Step 1: Understand the Problem

We need to find the first five numbers that are multiples of both 4 and 5. These are called common multiples.

Step 2: Find the Least Common Multiple (LCM)

First, determine the least common multiple of 4 and 5. The LCM is the smallest number that both 4 and 5 divide evenly into. Since 4 and 5 are coprime (they have no common prime factors), the LCM is the product of both numbers: \[ LCM(4, 5) = 4 \times 5 = 20 \]

Step 3: Generate Common Multiples

Once the LCM (20) is found, the common multiples are simply multiples of the LCM. Generate the first five multiples of 20 by multiplying 20 by 1, 2, 3, 4, and 5: \[ 20 \times 1 = 20, \quad 20 \times 2 = 40, \quad 20 \times 3 = 60, \quad 20 \times 4 = 80, \quad 20 \times 5 = 100 \]

Step 4: List the Common Multiples

List out the first five common multiples which we calculated: 20, 40, 60, 80, and 100.

Key Concepts

Understanding the Least Common Multiple (LCM)

The least common multiple, often abbreviated as LCM, is an important concept in mathematics. It finds the smallest multiple that is evenly divisible by two or more numbers.
In simpler terms, if you have two numbers, the LCM is the smallest number that both of these numbers can divide into without leaving a remainder.
For example, when finding the LCM of 4 and 5, you want the smallest number that both numbers divide into perfectly.
4 divides into 20 exactly 5 times (20 ÷ 4 = 5), and 5 divides into 20 exactly 4 times (20 ÷ 5 = 4).
Since 4 and 5 have no common factors (other than 1), their LCM can be discovered by simply multiplying them together:

• LCM(4, 5) = 4 × 5 = 20.

The result, 20, is the LCM. This forms a foundational step for finding common multiples.

Exploring Coprime Numbers

Coprime numbers are numbers that do not share any prime factors, except for 1.
In other words, two numbers are coprime if their greatest common divisor (GCD) is 1.
When two numbers are coprime, it means they line up in a unique way that makes them particularly easy to work with when finding their least common multiple.
For instance, since 4 and 5 are coprime, they have no factors in common.
This simplifies the process of finding their LCM, as you only need to multiply the numbers together to find it.
This is a handy property as it speeds up calculations, especially when solving mathematical problems involving LCM and common multiples.

• Coprime relationship: Helps in calculating the LCM faster.
• Example of coprime numbers: 4 and 5.

This is why understanding coprimes is crucial when solving problems on common multiples.

The Concept of Multiples

A multiple of a number is what you get when you multiply that number by an integer.
Multiples are like steps on a ladder, increasing by the same amount with each step.
For example, multiples of 4 include 4, 8, 12, 16, and so on, which you get by multiplying 4 by 1, 2, 3, 4, etc.
Similarly, for 5, the multiples would be 5, 10, 15, 20, etc.
When we focus on finding common multiples of two numbers, we're looking for numbers that appear in both lists of multiples.

• Multiples provide a quick and systematic way to understand number patterns.
• Helps in identifying aligned values when solving mathematical equations or problems.

By understanding multiples, you can quickly identify common multiples, providing a clear path to solve for the first few common multiples.

Tackling Mathematics Problem Solving

Mathematics problem solving often involves analyzing a problem, breaking it into smaller parts, and understanding the underlying concepts.
Finding common multiples, like in the exercise on 4 and 5, is a process that applies this problem-solving approach.
Here's a step-by-step breakdown on how you approach such a problem:

• Understand the problem: Figure out what exactly is being asked (i.e., Find common multiples).
• Identify the key concept: Recognize concepts like LCM and coprime numbers that are crucial to solving the problem.
• Engage in systematic steps: Follow logic and reason to deduce each step, ensuring no detail is missed out.

Such proficiency in problem solving translates well across various areas of mathematics, not just with multiples and divisions.
Equipped with these strategies, students can tackle a wide range of mathematical problems effectively.