I actually got #9 correct (after thinking about it for hours and writing it on my whiteboard multiple times, and maybe writing a python program to do it for me)

My method was setting X and Y to be the base-5 and base-6 counterparts of N, respectively, and setting

N = abc (base 10)

X = defgh (base 5)

Y = jkmn (base 6)

Where the letters denote digits. (i.e. abc could be 243 or 398)

Then from here, remember that

X = 625d + 125e + 25f + 5g + h in decimal

Try converting all of them to decimal.  Remember N=X=Y in decimal!

Try getting an equation that relates c, h, and n, and then use the fact that the last 2 digits of S = the last 2 digits of N.

There are probably better solutions.

This is Charles Zhang, signing off for now.