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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
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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.?.
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