I realised almost as soon as I posted you didn't actually have to establish an upper bound (written in my working out but not strictly necessary, could just skip to the exact step where you found the answer).

You have a possible answer, but you can do a lot better than that -- so here's my working out from before. 2003 doesn't have to be in January, which is what you've got there.

In the interest of a short explanation (and you can have a look at this on your own -- I think deriving this for yourself is good practice because you've overshot by a long way in your solution), for these two years to be in the same converted month, 2003 must be no later than March. Hence, we want 2003 to be as late as possible on the 31st of March which happens to be the 90th day of the year ie. since the days get rounded down, we effectively want to be as close to 91 as possible. Our hypothetical year \(X\) is thus equal to \(\lim_{x \to 91} \frac{2003 \times 365}{x} \approx 8034\).

To verify that 1485 is in the same month, we can sub in \(\frac{1485 \times 365}{8034} = 67.47\), which so happens to be March 8.