1. Prove that the product of two odd integers is odd.
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
Problem 1PE
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![**1. Prove that the product of two odd integers is odd.**
To prove this, let's consider two odd integers. By definition, an odd integer can be expressed in the form \(2k + 1\), where \(k\) is an integer.
Let the two odd integers be \(a = 2m + 1\) and \(b = 2n + 1\), where \(m\) and \(n\) are integers.
The product of \(a\) and \(b\) is:
\[
ab = (2m + 1)(2n + 1)
\]
Expanding this expression:
\[
ab = 4mn + 2m + 2n + 1
\]
\[
ab = 2(2mn + m + n) + 1
\]
Since \(2mn + m + n\) is an integer (as the sum and product of integers are integers), we can see that:
\[
ab = 2k + 1
\]
where \(k\) is the integer \(2mn + m + n\).
Therefore, \(ab\) is of the form \(2k + 1\), which means it is odd. This concludes the proof that the product of two odd integers is odd.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F44c3b237-e4ca-4e38-a278-251c2495f1b6%2F859a9000-52d9-4b56-8d15-1dbea383ca45%2Fufp3udh_processed.png&w=3840&q=75)
Transcribed Image Text:**1. Prove that the product of two odd integers is odd.**
To prove this, let's consider two odd integers. By definition, an odd integer can be expressed in the form \(2k + 1\), where \(k\) is an integer.
Let the two odd integers be \(a = 2m + 1\) and \(b = 2n + 1\), where \(m\) and \(n\) are integers.
The product of \(a\) and \(b\) is:
\[
ab = (2m + 1)(2n + 1)
\]
Expanding this expression:
\[
ab = 4mn + 2m + 2n + 1
\]
\[
ab = 2(2mn + m + n) + 1
\]
Since \(2mn + m + n\) is an integer (as the sum and product of integers are integers), we can see that:
\[
ab = 2k + 1
\]
where \(k\) is the integer \(2mn + m + n\).
Therefore, \(ab\) is of the form \(2k + 1\), which means it is odd. This concludes the proof that the product of two odd integers is odd.
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