Problem 147 Number Sense Explain why the pro... [FREE SOLUTION]

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Elementary Algebra Within Reach

Ron Larson, Kimberly Nolting

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 1: Problem 147

Number Sense Explain why the product of an even integer and any other integer is even. What can you conclude about the product of two odd integers?

Short Answer

Expert verified

The product of an even number and any other number is always an even number. This is because the product is expressed as a multiple of 2, which is the definition of even numbers. Similarly, the product of two odd numbers is an odd number because the product can be expressed in the form of an odd number.

Step by step solution

Understand Even and Odd Numbers

An even number can be defined as any integer that is a multiple of 2. In other words, an integer $x$ is even if it can be written in the form $2k$ where $k$ is an integer. An odd number is an integer which is not a multiple of 2, and can be expressed as $2k + 1$ where $k$ is an integer.

Proof that the Product of an Even and Any Integer is Even

Let's take an even number, represented as $2k$, where $k$ is an integer, and another integer let's denote it as $m$. The product of the even number $2k$ and an integer $m$ can be represented as $2k \cdot m = 2(km)$. This product is a multiple of 2 and by definition it is an even number.

Conclusion About the Product of two Odd Integers

An odd number can be expressed in the form $2k + 1$, where $k$ is an integer. If we take two odd numbers denoted as $2n + 1$ and $2m + 1$ respectively, the product of these two odd numbers is $(2n + 1)(2m + 1) = 4nm + 2n + 2m + 1 = 2(2nm + n + m) + 1$. This form is again in the form of an odd number $2k + 1$, where $k = 2nm + n + m$. Hence, the product of two odd numbers is also an odd number.

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91影视

Number Sense Explain why the product of an even integer and any other integer is even. What can you conclude about the product of two odd integers?

Short Answer

Step by step solution

Understand Even and Odd Numbers

Proof that the Product of an Even and Any Integer is Even

Conclusion About the Product of two Odd Integers

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