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## Homework Statement

Prove by induction that [itex] n (n^2 +5) [/itex] is divisible by 6 for all positive integers [itex] n [/itex]

## The Attempt at a Solution

Let [itex] f(n) = n (n^2 +5) [/itex]

[itex] f(1) = 6 [/itex]

So, true for [itex] n=1 [/itex]

Assume true for [itex] f(k) [/itex]

For [itex] n = k + 1 [/itex]:

[itex] f(k+1) = (k+1)[(k+1)^2 +5] [/itex]

[itex] f(k+1)-f(k) = (k+1)(k^2 +2k +6) - k^3 + 5k [/itex]

[itex] f(k+1)-f(k) = k^3 + 2k^2 + 6k +k^2 + 2k + 6 - k^3 -5k [/itex]

[itex] f(k+1)-f(k) = 3k^2 + 3k +6 [/itex]

I'm totally stuck from here. I was expecting f(k+1) - f(k) to be divisible by six, so then f(k+1) would be equal to the sum of two numbers divisible by six, which would show that f(k+1) is divisible by six. How can I show that final term is divisible by six? Or have I made a dumb mistake somewhere? Thanks.

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