Problem 4 - Fundamentals of Mathematics

Question

Find the first five multiples of the following numbers. 7

Step-by-Step Solution

Answer

The first five multiples of 7 are 7, 14, 21, 28, and 35.

Step 1: Understanding Multiples

Multiples of a number are obtained by multiplying the number by integers starting from 1 and going upwards. Therefore, to find the first five multiples of 7, we will multiply 7 by the numbers 1, 2, 3, 4, and 5.

Step 2: Calculate the 1st Multiple

To find the first multiple of 7, multiply 7 by 1: \[ 7 \times 1 = 7 \]

Step 3: Calculate the 2nd Multiple

To find the second multiple of 7, multiply 7 by 2: \[ 7 \times 2 = 14 \]

Step 4: Calculate the 3rd Multiple

To find the third multiple of 7, multiply 7 by 3: \[ 7 \times 3 = 21 \]

Step 5: Calculate the 4th Multiple

To find the fourth multiple of 7, multiply 7 by 4: \[ 7 \times 4 = 28 \]

Step 6: Calculate the 5th Multiple

To find the fifth multiple of 7, multiply 7 by 5: \[ 7 \times 5 = 35 \]

Step 7: Conclusion

The first five multiples of 7 are 7, 14, 21, 28, and 35.

Key Concepts

Multiplication

Multiplication is one of the basic operations in mathematics. It involves the process of adding a number to itself a certain number of times. For example, multiplying 7 by 3 means adding 7 three times: 7 + 7 + 7.
It is commonly represented by the symbol "\( \times \)". This can be visualized as finding the total number of items when there are several groups, each containing a certain number of items.

• An example would be: If there are 3 baskets, each with 7 apples, multiplying gives you the total number of apples: 3 times 7 equals 21 apples.
• Another simple example: 5 times 4 equals 20.

Understanding multiplication is crucial as it forms the foundation for more advanced mathematical concepts. It's essential for solving problems related to area, volume, and even algebraic equations.

Integer Multiplication

Integer multiplication specifically deals with multiplying whole numbers without fractions or decimals. Integers include positive numbers, negative numbers, and zero.

• Positive Numbers: These are natural numbers like 1, 2, 3, etc.
• Negative Numbers: These numbers are less than zero, such as -1, -2, -3, and so on.
• Zero: This is a neutral integer with a special rule—any number multiplied by zero will always be zero.

When multiplying two integers, the sign of the result depends on the signs of the numbers being multiplied. This can be summarized as:

• Multiplying two positive integers results in a positive product (e.g. \(3 \times 4 = 12\)).
• Multiplying two negative integers also results in a positive product (e.g. \((-2) \times (-4) = 8\)).
• Multiplying a positive integer and a negative integer results in a negative product (e.g. \(5 \times (-3) = -15\)).

Integer multiplication is important in real-world applications like banking, coding, and statistical analysis.

Sequences

In mathematics, a sequence is an ordered list of numbers. Each number in a sequence is called a term. Sequences can follow specific patterns or rules which determine the values of the terms.
When dealing with multiples, you are actually working with a specific type of sequence. For example, the sequence of the first five multiples of 7 is: 7, 14, 21, 28, 35.
This sequence follows a simple rule: start with 7 and repeatedly add 7.

• The first term: 7 \( (7 \times 1) \)
• The second term: 14 \( (7 \times 2) \)
• The third term: 21 \( (7 \times 3) \)
• The fourth term: 28 \( (7 \times 4) \)
• The fifth term: 35 \( (7 \times 5) \)

Sequences are used to understand patterns and predict future events. In math, they can also lead into understanding series, calculus, and more complex algebraic functions.