When using math induction to prove that (n^3 - n + 3) is divisible by 3 for all natural numbers. What equation would you need to show is divisible by 3?

Question

anonymous · Answer

you can do this easier without induction by factoring

anonymous · Answer

since $x^3-n+3=(n-1)\times n \times (n+1)+3$ 
since one of the three consecutive integers $n-1, n, n+1$ is divisible by 3 and 3 is also

anonymous · Answer

or you can proceed by induction if you have to 
show it is true if $n=1$ and then assume it is true for $k$ i.e. assume 
$$k^3-k+3$$ is divisible by 3, then prove if that is the case the 
$$(k+1)^3-(k+1)+3$$ is also divisible by 3
it is a bunch of algebra to find the hidden $k^3-k+3$ in $(k+1)^3-(k+1)+3$

anonymous · Answer

Thank you!