Problem 8 - Fundamentals of Mathematics

Question

Find the first five common multiples of the following numbers. 2 and 5

Step-by-Step Solution

Answer

The first five common multiples of 2 and 5 are: 10, 20, 30, 40, and 50.

Step 1: Understanding Common Multiples

Common multiples of two numbers are those multiples that are shared by both numbers. To find these for 2 and 5, we list some of the multiples of each and then identify those that appear in both lists.

Step 2: Finding Multiples of 2

Multiples of 2 are found by multiplying 2 by whole numbers. The first few are: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, and so on.

Step 3: Finding Multiples of 5

Similarly, multiples of 5 are found by multiplying 5 by whole numbers. The first few are: 5, 10, 15, 20, 25, 30, and so on.

Step 4: Identifying Common Multiples

Now, we identify numbers that appear in both lists of multiples. From our lists, we see that 10, 20, 30, 40, etc., are in both.

Step 5: Listing the First Five Common Multiples

From our identification in the previous step, the first five common multiples of 2 and 5 are: 10, 20, 30, 40, and 50.

Key Concepts

Multiples of 2

Multiples of 2 are numbers that result from multiplying the number 2 by whole numbers. For instance, you would multiply 2 by 0, 1, 2, 3, and so on. This creates a series of numbers like 2, 4, 6, 8, 10, and continues infinitely.

• Start with 2, since 2 multiplied by 1 is 2.
• Each subsequent multiple adds 2 more to the last multiple.
• This creates a predictable pattern: 2, 4, 6, 8, 10, etc.

Understanding these multiples is crucial when trying to find common multiples shared with another number.

Multiples of 5

Multiples of 5 follow a similar rule. To find them, multiply 5 by whole numbers. This produces numbers like 5, 10, 15, 20, and so on.

• The first multiple, 5, is achieved by 5 multiplied by 1.
• Each new multiple is 5 more than the previous one.
• This forms a clear pattern: 5, 10, 15, 20, etc.

Recognizing such sequences can help when listing common multiples with another number.

Step by Step Solution

Finding common multiples requires comparing the multiples of two numbers. Let's break down the steps:

Step 1: Listing the Multiples

Starting with multiples of 2 and 5, as outlined above, you create two sets of numbers.

Step 2: Identifying Shared Numbers

Find numbers that appear in both lists. For 2 and 5, such numbers include 10, 20, 30, and so forth.

Step 3: Listing the First Five Common Multiples

From the identified numbers, pick the first five. Here, they are 10, 20, 30, 40, and 50.
This organized approach ensures the thorough and accurate determination of common multiples, helping with tasks like finding the least common multiple or planning schedules.