Induction · Challenge problem

"Claim": $5n = 0$ for every integer $n \ge 0$. "Proof" by strong induction: $P(0)$ holds. For $k + 1$, write $k + 1 = i + j$ with $0 \le i, j \le k$; by hypothesis $5i = 0$ and $5j = 0$, so $5(k + 1) = 0$. The step works for every $k \ge 0$ except one. Which $k$?