Do you need to use different resistors against each of the colours on an RGB LED?
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Yes, you need a separate resistor per LED. (or per color of an RGB LED, which just just 3 LEDs in one package)
Due to the physics of the LED, different colored LEDs have different "forward voltage"s (a primary characteristic of an LED). This site's LED color chart gives the forward voltages for their LEDs, but it does really depend on the LED in question. In general, the higher in frequency light an LED makes (the bluer it is), the higher the forward voltage. Often, a red LED's forward voltage is ~2V, a green one ~3V, and a blue one is ~3.4V, but it really does depend on the LED manufacturer and the exact frequency of the light emitted.
Once you know the forward voltage of an LED, you can use Ohm's Law to calculate the resistor you need for a given power supply voltage. Or you can use a handy LED calculator to help.