How much current is needed to light a bulb with 180 ohms of resistance using 120 volt power source?

V = I x R so current I = 1/2 amp. I bet the bulb is rated at 60 W because Watts = Current x Voltage.

Where V = voltage (volt)

I = current (ampere)

R = resistance (ohm)

Your question isn't easy to answer. A lamp has two 'resistances': a 'cold' resistance, and a 'hot' resistance. Before it is energised, it is cold, so its resistance is low; when it is energised, it becomes very hot, and its resistance increases significantly. So, the question is whether your '240 ohms' is the cold resistance or the hot resistance. If it is the cold resistance, then a current of 0.5 A will flow through it for a fraction of a second, then its resistance will increase significantly, and the current will fall to a very much smaller value.