What would cause a downstairs toilet to start running when an upstairs toilet is flushed both toilets keep running until a faucet or the washing machine is turned on?

the most likely answer to this question is a faulty washer, the reason it keeps running until the washing machine is turned on is cause the pressure is being withdrawn from the toilet. replace the washers to the toilets and monitor. if this fails you may have to consider replacing all the mechinisims inside