Answer
See explanations.
Work Step by Step
Step 1. Prove the statement is true when $n=1$: $1^3-1+3=3$ is divisible by 3, thus it is true for $n=1$.
Step 2. Assume the statement is true when $n=k$: we have $k^3-k+3$ is divisible by 3.
Step 3. Prove it is true for $n=k+1$: $(k+1)^3-(k+1)+3=k^3+3k^2+3k+1-k-1+3=(k^3-k+3)+3(k^2+k)$. As both $k^3-k+3$ and $3(k^2+k)$ are divisible by 3, their sum is also divisible by 3.
Thus, the statement is also true for $n=k+1$
Step 4. Conclusion: with mathematical induction, we have proved that the statement is true for all natural numbers $n$.
