Supplying 5V to 1.8V, 5mm (red)LED along with 74LS08 AND gate. What should be the value of resistance?

it not so much the voltage that you want to control, as the current. those LED's generally can be powered with as little as 1.5ma, but would be very dim. 7 or 8 ma would be plenty.

So, first, the resistor would be in series. connect the output of the AND gate to the cathode of the LED. connect the anode to one side of the resistor, and the other side to the 5v. now, when the output of the AND gate goes low (0v), the LED turns on. the reason for connecting it this way is IC's (in general) can sink more current than they can source.

now, to calculate the resistor size, you just use ohm's law R = E/I. E is 5v, I is anywhere between 1.5ma and 7ma (however bright you want it), but say, start at 4ma. to apply the numbers is just 5/.004, which = 1250 ohms. you will have a hard time finding a 1250 ohm resistor, so just use one that is close, like 1470 ohm (1.47k ohm) is standard.

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