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Chapter 5: Problem 99

If \(a\) is a factor of \(c\), what does this mean?

Short Answer

Expert verified
A number \(a\) is a factor of \(c\) if \(c\) can be divided by \(a\) without leaving a remainder. In other words, if there exists an integer \(b\) such that \(a * b = c\).

Step by step solution

01

Definition of a Factor

A number \(a\) is said to be a factor of another number \(c\) if \(c\) can be exactly divided by \(a\) without leaving any remainder. This is equivalent to saying there exists some integer \(b\) such that when \(a\) is multiplied by \(b\), the result is \(c\). In terms of an equation, this can be written as \(a * b = c\).
02

Example of a Factor

For instance, let's consider the numbers 2 and 8. Here, 2 can divide 8 exactly four times with no remainder. Hence, we can say that 2 is a factor of 8. It can be written using the equation above as \(2 * 4 = 8\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Division
Division is a basic mathematical operation where you determine how many times one number can be contained within another. In simpler terms, it's like sharing equally. When you divide a number, called the dividend, by another number, known as the divisor, you obtain a quotient.
Imagine you have 8 candies and you want to distribute them equally among 4 friends. Here, 8 is the dividend, 4 is the divisor, and the result, 2, is the quotient, because each friend will get 2 candies.
Division plays a crucial role in identifying factors of a number since factors imply that division can occur without any leftovers.
What is an Integer?
Integers are whole numbers that can be positive, negative, or zero. They do not include fractions or decimals. Examples of integers include -3, 0, 1, and 7.
In mathematics, integers are denoted by the symbol \( \mathbb{Z} \).
Understanding integers is essential when dealing with factors because both the factor and the result after division (quotient) should be whole numbers.
When you say a number \(a\) is a factor of \(c\), it implies that when \(c\) is divided by \(a\), the result ought to be an integer.
Exploring Multiplication
Multiplication is the mathematical operation of scaling one number by another. It's often viewed as repeated addition. For instance, multiplying 3 by 4 (\(3 \times 4\)) is akin to adding 3 four times: \(3 + 3 + 3 + 3 = 12\).
Multiplication is closely linked to factors. When \(a\) is a factor of \(c\), there exists another integer \(b\) such that \(a \times b = c\). In this relationship, \(a\) and \(b\) are viewed as building blocks that, when multiplied, reconstruct \(c\).
Understanding multiplication helps clarify why some numbers evenly divide others, showcasing the interconnectedness of multiplication and division.
Understanding Equations
An equation is a statement that asserts the equality of two expressions, typically involving variables and numbers. Equations are key in problem-solving and understanding mathematical relationships.
In the context of factors, the equation \(a \times b = c\) illustrates how factors work. It says that when you multiply \(a\), the factor, by another integer \(b\), you get \(c\).
Equations are fundamental tools in mathematics, providing a framework to express real-world problems and solutions in a clear, concise manner.
The Concept of Remainder
The remainder is the portion left over after division when a number doesn't divide evenly. If you divide 9 by 4, you get 2 with a remainder of 1, because 4 goes into 9 twice, leaving 1.
When a number is a factor of another, the remainder of their division is zero. This means that factors characterize numbers that "fit" perfectly into others without any leftover parts.
Knowing about remainders is vital when exploring factors since it highlights whether or not a number divides evenly into another, confirming its status as a factor.

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Most popular questions from this chapter

Write the first six terms of the arithmetic sequence with the first term, \(a_{1}\), and common difference, \(d\). \(a_{1}=4.5, d=-0.75\)

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