UPSC Exam > UPSC Questions > Show that the square of any positive integer ...
Start Learning for Free
Show that the square of any positive integer is either of the form 4q or 4q+1 for some integer?
Most Upvoted Answer
Show that the square of any positive integer is either of the form 4q ...
Proof:
1. Square of an Even Integer:
When we square an even integer, say 2n, we get (2n)^2 = 4n^2. This is clearly of the form 4q where q = n^2.
2. Square of an Odd Integer:
When we square an odd integer, say 2n+1, we get (2n+1)^2 = 4n^2 + 4n + 1 = 4(n^2 + n) + 1. This is of the form 4q+1 where q = n^2 + n.
Conclusion:
Therefore, the square of any positive integer can be represented in the form 4q or 4q+1 for some integer q. This is true for both even and odd integers.
This proof shows that any positive integer when squared will result in a number that is either of the form 4q or 4q+1. The key idea is to consider the cases of even and odd integers separately and demonstrate that the resulting square follows the specified forms. This proof provides a clear and concise explanation of the pattern observed in the squares of positive integers.
1. Square of an Even Integer:
When we square an even integer, say 2n, we get (2n)^2 = 4n^2. This is clearly of the form 4q where q = n^2.
2. Square of an Odd Integer:
When we square an odd integer, say 2n+1, we get (2n+1)^2 = 4n^2 + 4n + 1 = 4(n^2 + n) + 1. This is of the form 4q+1 where q = n^2 + n.
Conclusion:
Therefore, the square of any positive integer can be represented in the form 4q or 4q+1 for some integer q. This is true for both even and odd integers.
This proof shows that any positive integer when squared will result in a number that is either of the form 4q or 4q+1. The key idea is to consider the cases of even and odd integers separately and demonstrate that the resulting square follows the specified forms. This proof provides a clear and concise explanation of the pattern observed in the squares of positive integers.
|
Explore Courses for UPSC exam
|
|
Similar UPSC Doubts
Top Courses for UPSCView all
Show that the square of any positive integer is either of the form 4q or 4q+1 for some integer?
Question Description
Show that the square of any positive integer is either of the form 4q or 4q+1 for some integer? for UPSC 2025 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Show that the square of any positive integer is either of the form 4q or 4q+1 for some integer? covers all topics & solutions for UPSC 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Show that the square of any positive integer is either of the form 4q or 4q+1 for some integer?.
Show that the square of any positive integer is either of the form 4q or 4q+1 for some integer? for UPSC 2025 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Show that the square of any positive integer is either of the form 4q or 4q+1 for some integer? covers all topics & solutions for UPSC 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Show that the square of any positive integer is either of the form 4q or 4q+1 for some integer?.
Solutions for Show that the square of any positive integer is either of the form 4q or 4q+1 for some integer? in English & in Hindi are available as part of our courses for UPSC.
Download more important topics, notes, lectures and mock test series for UPSC Exam by signing up for free.
Here you can find the meaning of Show that the square of any positive integer is either of the form 4q or 4q+1 for some integer? defined & explained in the simplest way possible. Besides giving the explanation of
Show that the square of any positive integer is either of the form 4q or 4q+1 for some integer?, a detailed solution for Show that the square of any positive integer is either of the form 4q or 4q+1 for some integer? has been provided alongside types of Show that the square of any positive integer is either of the form 4q or 4q+1 for some integer? theory, EduRev gives you an
ample number of questions to practice Show that the square of any positive integer is either of the form 4q or 4q+1 for some integer? tests, examples and also practice UPSC tests.
|
|
Explore Courses for UPSC exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup