Class 10 Exam > Class 10 Questions > For any positive integer n, prove that n^3-n ...
Start Learning for Free
For any positive integer n, prove that n^3-n is divisible by 6.?
Most Upvoted Answer
For any positive integer n, prove that n^3-n is divisible by 6.?
Let :-
a = n³ - n
= n(n² - 1)
= n(n - 1)(n + 1)
= (n - 1)n (n + 1)
1) Now out of three (n - 1),n and (n + 1) one must be even so a is divisible by 2
2) Also (n - 1),n and (n + 1) are three consecutive integers thus as proved a must be divisible by 3
From (1) and (2)
a must be divisible by 2 × 3 = 6
Hence n³ - n is divisible by 6 for any positive integer n.
Hope this helps u...
a = n³ - n
= n(n² - 1)
= n(n - 1)(n + 1)
= (n - 1)n (n + 1)
1) Now out of three (n - 1),n and (n + 1) one must be even so a is divisible by 2
2) Also (n - 1),n and (n + 1) are three consecutive integers thus as proved a must be divisible by 3
From (1) and (2)
a must be divisible by 2 × 3 = 6
Hence n³ - n is divisible by 6 for any positive integer n.
Hope this helps u...
Community Answer
For any positive integer n, prove that n^3-n is divisible by 6.?
Answern³ - n = n(n²-1) = n(n -1)(n + 1) is divided by 3 then possible reminder is 0, 1 and 2 [ ∵ if P = ab + r , then 0 ≤ r < a="" by="" euclid="" lemma="" ]="" />∴ Let n = 3r , 3r +1 , 3r + 2 , where r is an integer Case 1 :- when n = 3r Then, n³ - n is divisible by 3 [∵n³ - n = n(n-1)(n+1) = 3r(3r-1)(3r+1) , clearly shown it is divisible by 3 ] Case2 :- when n = 3r + 1 e.g., n - 1 = 3r +1 - 1 = 3r Then, n³ - n = (3r + 1)(3r)(3r + 2) , it is divisible by 3 Case 3:- when n = 3r - 1 e.g., n + 1 = 3r - 1 + 1 = 3r Then, n³ - n = (3r -1)(3r -2)(3r) , it is divisible by 3 From above explanation we observed n³ - n is divisible by 3 , where n is any positive integers
Attention Class 10 Students!
To make sure you are not studying endlessly, EduRev has designed Class 10 study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in Class 10.
|
Explore Courses for Class 10 exam
|
|
Similar Class 10 Doubts
Top Courses for Class 10View all
For any positive integer n, prove that n^3-n is divisible by 6.?
Question Description
For any positive integer n, prove that n^3-n is divisible by 6.? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about For any positive integer n, prove that n^3-n is divisible by 6.? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for For any positive integer n, prove that n^3-n is divisible by 6.?.
For any positive integer n, prove that n^3-n is divisible by 6.? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about For any positive integer n, prove that n^3-n is divisible by 6.? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for For any positive integer n, prove that n^3-n is divisible by 6.?.
Solutions for For any positive integer n, prove that n^3-n is divisible by 6.? in English & in Hindi are available as part of our courses for Class 10.
Download more important topics, notes, lectures and mock test series for Class 10 Exam by signing up for free.
Here you can find the meaning of For any positive integer n, prove that n^3-n is divisible by 6.? defined & explained in the simplest way possible. Besides giving the explanation of
For any positive integer n, prove that n^3-n is divisible by 6.?, a detailed solution for For any positive integer n, prove that n^3-n is divisible by 6.? has been provided alongside types of For any positive integer n, prove that n^3-n is divisible by 6.? theory, EduRev gives you an
ample number of questions to practice For any positive integer n, prove that n^3-n is divisible by 6.? tests, examples and also practice Class 10 tests.
|
|
Explore Courses for Class 10 exam
|
|
Suggested Free Tests
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