Problem 6 - Fundamentals of Mathematics

Question

Find the first five common multiples of the following numbers. 2 and 4

Step-by-Step Solution

Answer

The first five common multiples of 2 and 4 are 4, 8, 12, 16, and 20.

Step 1: Understand the Definition of Common Multiples

A common multiple of two or more numbers is a number that is a multiple of each of the numbers. To find the first five common multiples of 2 and 4, we will identify the multiples of each number and then find the numbers that appear in both lists.

Step 2: List the Multiples of 2

Multiples of 2 are the numbers you get when 2 is multiplied by an integer. The first few multiples of 2 are: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, etc.

Step 3: List the Multiples of 4

Multiples of 4 are the numbers you get when 4 is multiplied by an integer. The first few multiples of 4 are: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, etc.

Step 4: Identify the Common Multiples

Compare the multiples of 2 and 4. The numbers that are present in both lists are the common multiples. From the lists we made, the common multiples are: 4, 8, 12, 16, 20.

Step 5: Verify the Common Multiples

Confirm by checking if these common multiples can be divided evenly by both 2 and 4. For example, 4 ÷ 2 = 2 and 4 ÷ 4 = 1, and similarly for 8, 12, 16, and 20, ensuring they are true common multiples.

Key Concepts

Understanding Multiples of Integers

To comprehend the idea of multiples, it's essential to know that a multiple of an integer is what you get when you multiply that integer by any whole number. For example, if we take the integer 2, its multiples would be 2, 4, 6, 8, and so on, because these numbers are results of multiplying 2 by 1, 2, 3, 4, etc. Similarly, multiples of 3 would be 3, 6, 9, etc. The list of multiples extends infinitely as you can multiply by increasingly larger numbers.

When dealing with multiple numbers, you'll often need to find numbers they all multiply into, known as common multiples. This is helpful in solving different math problems such as finding periods when events coincide or synchronizing transactions. The easiest common multiple to spot is the Least Common Multiple (LCM), which is simply the smallest among them. Understanding multiples is foundational in tackling more complex mathematics problems.

Solving Mathematics Problems Involving Common Multiples

When addressing mathematics problems involving common multiples, start by listing the multiples of each number. This systematic approach helps in ensuring that you don't miss any potential common multiples. Once you have your lists ready, compare them to find numbers appearing in both lists—these are your common multiples.

For instance, to find the common multiples of 2 and 4, begin by writing out their multiples. Multiples of 2 would be 2, 4, 6, and so on, while multiples of 4 would be 4, 8, 12, etc. Now, observe the lists: 4 appears in both, so it is a common multiple, and so do 8, 12, and 16.

This step-by-step methodical way helps in breaking down seemingly complex math problems into manageable tasks. Listing out multiples not only aids in finding common multiples but also provides insight into the relationships between numbers and enhances understanding of division and multiplication.

The Role of Division in Identifying Common Multiples

Division plays a crucial role in verifying common multiples. To confirm if a number is a common multiple of two numbers, it must be divisible by each of the given numbers without leaving a remainder. For example, if you suspect a number is a common multiple of 2 and 4, divide it by 2 and then by 4. If both divisions result in whole numbers, the number is indeed a common multiple.

Take 8 as an example:

• Dividing 8 by 2 gives 4
• Dividing 8 by 4 gives 2

Both results are integers, which certifies 8 is a common multiple of 2 and 4. This check ensures that all numbers identified as common multiples are valid, improving accuracy in mathematical calculations. Proper use of division in this context strengthens the reliability of solutions across various mathematical problems.