Problem 1 - Fundamentals of Mathematics

Question

Find the first five multiples of the following numbers. 4

Step-by-Step Solution

Answer

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

Step 1: Understand Multiples

Multiples of a number are the results of multiplying that number by an integer. For example, multiples of 4 are obtained by multiplying 4 with 1, 2, 3, and so on.

Step 2: Calculate First Multiple

To find the first multiple, multiply 4 by 1. \[4 imes 1 = 4\] So, the first multiple of 4 is 4.

Step 3: Calculate Second Multiple

Multiply 4 by 2 to get the second multiple. \[4 imes 2 = 8\] Thus, the second multiple of 4 is 8.

Step 4: Calculate Third Multiple

Now, multiply 4 by 3 for the third multiple. \[4 imes 3 = 12\] Consequently, the third multiple of 4 is 12.

Step 5: Calculate Fourth Multiple

Multiply 4 by 4 to find the fourth multiple. \[4 imes 4 = 16\] Therefore, the fourth multiple of 4 is 16.

Step 6: Calculate Fifth Multiple

Finally, multiply 4 by 5 to obtain the fifth multiple. \[4 imes 5 = 20\] Hence, the fifth multiple of 4 is 20.

Key Concepts

Number Multiples

Understanding multiples is a fundamental concept in mathematics. Multiples of a number emerge when you multiply that number by any whole number, also known as an integer. For example, if you take the number 4 and multiply it by integers like 1, 2, 3, and so on, you obtain multiples of 4.

• Multiples of 4 include 4, 8, 12, 16, and 20.
• They extend indefinitely as you continue multiplying by higher integers.

The sequence of multiples forms an arithmetic sequence, where each term after the first is generated by adding the same amount to the previous term.

Integer Multiplication

Integer multiplication involves taking two whole numbers (without fractions) and calculating their product. In the case of finding multiples, you multiply a specific integer by other integers to create a series of results.
When calculating multiples, such as those for the number 4, you perform several integer multiplications:

• 4 \( \times \) 1 = 4
• 4 \( \times \) 2 = 8
• 4 \( \times \) 3 = 12
• 4 \( \times \) 4 = 16
• 4 \( \times \) 5 = 20

These examples demonstrate integer multiplication by iterating multiplication through successive integers.

Basic Algebra

In basic algebra, understanding how to multiply numbers is crucial for solving various problems. The principles of equality, where both sides of an equation balance, often rely on being able to multiply and find multiples.
In our exercise, we could express finding multiples of 4 using a basic algebraic expression:

• Let \( n \) be any integer.
• The expression for a multiple of 4 is \( 4 \times n \).

Using algebra allows you to generalize this concept for any number, not just 4, providing a clear framework for understanding number relationships.

Arithmetic Sequences

An arithmetic sequence is a series of numbers in which the difference between consecutive terms is constant. For the multiples of a number, this difference is equal to the number itself. Each term in the arithmetic sequence for multiples of 4 is obtained by adding 4 to the previous term.
For example, 4, 8, 12, 16, and 20 form an arithmetic sequence with:

• A common difference of 4.
• The formula for the \( n \)-th term \( a_n = 4 + (n-1) \times 4 \).

This relationship shows the consistency and predictability of arithmetic sequences in mathematics.