jmc

The remainder when a number is divided by 10 is simply the units digit of that number. So we only look for the units digit of the product. With $1734\times 5389$, $4\times 9=36$, so the result will have a units digit of 6. Then we multiply 6 by the units digit of $80,607$ and get $6\times 7=42$. That means the final product will have a units digit of $\boxed{2}$.