The question says $10\cdot n=0$ for all $n\in\mathbb{Z}$ with $n\geq 0$.

Here is my proof by strong induction:

Base case: $10\cdot0=0$.

Let $k\geq 0$, and suppose that for any $m\leq k$ we have that $10\cdot m=0$.

Consider $10\cdot(k+1)$. The number $k+1$ can be written as $m+l$ for some numbers $0\leq m,l\leq k$. By the induction hypothesis, $10\cdot m=0=10\cdot l$.

But then: $10\cdot(k+1)=10\cdot(m+l) = 10\cdot m+10\cdot l=0+0=0$.

Apparently something went wrong in my proof by strong induction, but I cannot seem to figure out what.