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Prove that n(n+1)(n+5) is a multiple of 3

 p(n)=n(n+1)(n+5)

p(1)=1(1+1)(1+5)=1x2x6=12

let p(k) b true

p(k)=k(k+1)(k+5)

on solving

=k3+6k2+5k=3y( as its a multiple of 3)

k3=3y-6k2-5k ------(1)

p(k+1)= k+1 ( k+1 +1) k+1 +5)

          =k+1 ( k+2) k+6)

on solving =k3+9k2+20k +12

putting value of k3 from (1)

=3y-6k2-5k+9k2+20k +12

=3y+3k2+15k +12

3(y+k25k +4)

therefore we can say p(k+1) is a multiple of 3

p(k) is true so p(n) is also true and its a multiple of 3

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Fantasic Shuvi Dobhai. ThankYou..
 
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there should be three cases
case (i)-when n is multiple of three
n=3k where k is a integer.
then n (n+1) (n+5) = 3k (3k+1) (3k+5)
it is divisible by three.
case(ii)-
when n is not multiple of three and n=3k+1;
the n(n+1)(n+5)= (3k+1) ((3k+4) (3k+6)
3(3k+1) (3k+4) (k+2)
it is divisible by three.
case(iii)-when n is  not multiple of three and n=3k+2
then n (n+1) (n+5)= (3k+2) (3k+3) (3k+7)
3  (3k+2) (k+1) (3k+7)
Hence it is divisible by 3
Hope you understands
 
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