J Rollinger
You are giving dangerous advice. Workers have died from a single summer job. But it does take years after exposure for the diseases to develop. Just because you did not develop it does not mean that others will not. The chances for dying from exposure is too great to take lightly. By your reasoning smoking has no potential for harm either. You were very lucky and foolish. Visit the families of the workers at Libby Montana to see that you do not have to be even a worker to be affected. Not every one that comes in contact with air borne asbestos gets the three main asbestos related diseases but who would want any of them, they are fatal.
On the other hand you must be exposed to it in the air. Tiles are a poor example of how safe it is as unless you take a grinder to them and grind them down they pose no threat. Most people do not grind floor tiles. Asbestos encapsulated or embedded in a material and not being disturbed are generally very safe as there is no process to get the fibres air borne.
My job includes managing the asbestos on a military base and although I have never been an asbestos worker, I have taken the asbestos awarness course and the one which has certified me to be an asbestos remover. I simply sample materials that may contain asbestos.If you actually read the information about asbestos you will see that making it airborne is NOT safe.
If you think the washer is asbestos and wish to remove it, spray it lightly with water before you do so and remove it and place it in a plastic bag just as Kirk says. Wash your hands aftgerwards or wear gloves and wash them. When I first started my current position I submitted a work order to remove the asbestos in my former workplace, floor tile. The tile surface was wearing out and actually had a hole worn through. At the same time I did not recommend replacing the same tile in the next room as it as a storage room and the tiles are in excellent condition. In the first case the floor tile presented a danger to the worker in the room ( a photographer) and even a bigger threat to the cleaners but the second room, a photo storage posed no threat even though both had the same flooring.
Most asbestos is only dangerous when disturbed and we enounter it in our daily lives at home or at work but when it is disturbed to be removed, the area must be sealed and workers protected. Wearing an air filter will be added protection.
It may take 30 years to develop asbestos related diseases but it can stem from a short period of exposure like a summer job. Also being a smoker greatly increases your chances to get those diseases. Comparisions to exposure to other pollulants are misleading, some elements take many doses whereas others a single exposure.
That all said I would remove and replace the washer if it was coming apart and I would keep it if it was totally intact.