what happens if I put the resistor on the cathode instead of using 3 resistors for each color anode?
Then it won't work. Once you put the red LED on then the others won't come on because the red LED clamps the voltage at a lower value than is needed to turn on the other colours.
Why? Even for a red LED consuming 25 mA, the required resistor for running it on a 5V line(that's what Arduino provides, right?) is only 120 Ohms.
Most LEDs work at 20mA not 25mA. Yes you can use a lower resistor but most LEDs are good enough at between 10 to 15mA