1. The problem statement, all variables and given/known data Prove by induction that [itex] n (n^2 +5) [/itex] is divisible by 6 for all positive integers [itex] n [/itex] 3. 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.