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prove that n 3 + 3n 2 + 5n + 3 is divisible by 3 for any natural number n

prove that n3 + 3n2 + 5n + 3 is divisible by 3  for any natural number n

Grade:11

2 Answers

Arun Kumar IIT Delhi
askIITians Faculty 256 Points
9 years ago
Hello Student,
n^3 + 3n^2 + 5n + 3
=n^3 + 3n^2 + 2n+3n + 3
=n(n^2 + 3n + 2)+3n + 3
=n(n+1)(n+2)+3n+3
Prooved
Thanks & Regards
Arun Kumar
Btech, IIT Delhi
Askiitians Faculty
mycroft holmes
272 Points
9 years ago
By Fermat’s theorem, n3-n is divisible by 3.
 
Hence n3+3n2+5n+3 = (n3-n)+(3n2+6n+3) = 3m+3(n+1)2 for some integer m making the expression obviously a multiple of 3.
 
 

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