This problem sucks!

Since $44 \equiv 8 \pmod{12}$ $113 \equiv 5 \pmod{12}$ Thus we know that $44 \cdot 113 \equiv 8 \cdot 5 \equiv 40 \equiv 4 \pmod{12},$ meaning there are 4 sodas leftover.

This problem doesn't suck! I liked it. First you simply calculate the total number of sodas, which is simply $113 \times 44=4972$ . We can evaluate $4972 \pmod{12}$ by recognizing that it can be factored as $(108+5) \times 44 \pmod{12}$ . When this is distributed, the $108 \times 44$ disappears because $108 \equiv 0 \pmod{12}$ . We are left with $4 \times 44 \pmod{12}$ . Doing likewise to $44$ , we can evaluate this to simply be $20 \pmod{12} \equiv \boxed{4}$ and we're done! :D

$\frac{113 \times 44}{12}=414 \frac{4}{12}$

It means that there are 414 cases and there are $\boxed{4}$ sodas left.