jmc

We will consider this number $\mathrm{mod}5$ and $\mathrm{mod}9$. By looking at the last digit, it is obvious that the number is $\equiv 4\mathrm{mod}5$. To calculate the number $\mathrm{mod}9$, note that $$123456\cdots 4344\equiv 1+2+3+4+5+6+7+8+9+(1+0)+(1+1)+\cdots +(4+3)+(4+4)\equiv 1+2+\cdots +44\mathrm{mod}9,$$ so it is equivalent to $$\frac{44\cdot 45}{2}=22\cdot 45\equiv 0\mathrm{mod}9.$$ Let $x$ be the remainder when this number is divided by $45$. We know that $x\equiv 0(\mathrm{mod}9)$ and $x\equiv 4(\mathrm{mod}5)$, so by the Chinese remainder theorem, since $9(-1)\equiv 1(\mathrm{mod}5)$, $x\equiv 5(0)+9(-1)(4)(\mathrm{mod}5\cdot 9)$, or $x\equiv -36\equiv \boxed{9}(\mathrm{mod}45)$.