Problem 21 - Basic College Mathematics

Question

Multiply by \(1,2,3,\) and so on, to find ten multiples of each number. $$ 3 $$

Step-by-Step Solution

Answer

The first ten multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, 27, and 30.

Step 1: Understand the Concept of Multiples

A multiple of a number is the product of that number and an integer. For example, the multiples of 3 are found by multiplying 3 by the integers 1, 2, 3, and so on.

Step 2: Multiply by 1

Start by multiplying 3 by 1 to get the first multiple. \( 3 \times 1 = 3 \)

Step 3: Multiply by 2

Next, multiply 3 by 2 to get the second multiple. \( 3 \times 2 = 6 \)

Step 4: Multiply by 3

Multiply 3 by 3 to get the third multiple. \( 3 \times 3 = 9 \)

Step 5: Multiply by 4

Multiply 3 by 4 to get the fourth multiple. \( 3 \times 4 = 12 \)

Step 6: Multiply by 5

Multiply 3 by 5 to get the fifth multiple. \( 3 \times 5 = 15 \)

Step 7: Multiply by 6

Multiply 3 by 6 to get the sixth multiple. \( 3 \times 6 = 18 \)

Step 8: Multiply by 7

Multiply 3 by 7 to get the seventh multiple. \( 3 \times 7 = 21 \)

Step 9: Multiply by 8

Multiply 3 by 8 to get the eighth multiple. \( 3 \times 8 = 24 \)

Step 10: Multiply by 9

Multiply 3 by 9 to get the ninth multiple. \( 3 \times 9 = 27 \)

Step 11: Multiply by 10

Finally, multiply 3 by 10 to get the tenth multiple. \( 3 \times 10 = 30 \)

Key Concepts

Multiplication

Multiplication is one of the basic operations in arithmetic. When we multiply, we add a number to itself a certain number of times. For example, to find the product of 3 and 2, you add 3 two times: 3 + 3 = 6. This can also be written as 3 x 2 = 6. Multiplication can be symbolized using different signs such as 'x', '*', or '·'. Understanding multiplication helps us solve many everyday problems, such as calculating total costs, distances, and much more.
When learning multiplication, it's useful to remember that:

• The order in which you multiply numbers does not change the result (commutative property): 3 x 2 is the same as 2 x 3.
• Multiplying any number by 1 leaves it unchanged: 3 x 1 = 3.
• Multiplying any number by 0 gives 0: 3 x 0 = 0.

Practicing with smaller numbers helps you get the hang of it. Once you're comfortable, you can easily multiply larger numbers too.

Basic Arithmetic

Basic arithmetic includes the fundamental operations: addition, subtraction, multiplication, and division. These operations are crucial for everyday calculations and problem-solving.
When it comes to multiplication, understanding basic arithmetic means knowing how to quickly and correctly find the product of two numbers. Let's break down the steps to make it simpler:

• Identify the numbers you want to multiply (these are called factors).
• Use addition as a way to understand multiplying: multiplying a number is like adding it several times.
• Practice through repetition, such as with multiplication tables.

These steps help you become more fluent in basic arithmetic, especially in multiplication. By mastering these elements, you'll find it much easier to approach more complex math concepts later on.

Integer Multiples

When we talk about multiples, we refer to the product of a number and an integer. An integer is a whole number, and it can either be positive or negative. But for simplicity, we often use positive integers when learning about multiples. So, a multiple of 3, for example, is any result you get by multiplying 3 by an integer: 3 x 1, 3 x 2, 3 x 3, etc.

• The first multiple of 3 is 3 x 1 = 3.
• The second multiple of 3 is 3 x 2 = 6.
• The third multiple of 3 is 3 x 3 = 9.

And we keep going! Multiples continue infinitely as long as you continue multiplying by integers. Knowing these can help solve many problems like finding common denominators or simplifying fractions. Remember that **every number is a multiple of itself**. It is also a helpful trick to remember that multiples of numbers grow predictably and systematically.