jmc

Let $N=109876543210$. Notice that $180=4\times 9\times 5$, so by the Chinese Remainder Theorem, it suffices to evaluate the remainders when $N$ is divided by each of $4$, $9$, and $5$. We can apply the divisibility rules to find each of these. Since the last two digits of $N$ are $10$, it follows that $N\equiv 10\equiv 2(\mathrm{mod}4)$. We know that $N$ is divisible by $5$, so $N\equiv 0(\mathrm{mod}5)$. Finally, since $N$ leaves the same residue modulo $9$ as the sum of its digits, then $$N\equiv 0+1+2+3+\cdots +9+1\equiv 1+\frac{9\cdot 10}{2}\equiv 46\equiv 1(\mathrm{mod}9).$$By the Chinese Remainder Theorem and inspection, it follows that $N\equiv 10(\mathrm{mod}4\cdot 9)$, and since $10$ is also divisible by $5$, then $N\equiv \boxed{10}(\mathrm{mod}180)$.