Connection between Main floor kitchen sink and basement kitchen sink.

About a month ago, we had finished painting and found our main floor kitchen sink clogged. While the work progressed, we used the basement kitchen to cook. With a help from a friend we were able to unclog the main floor kitchen drain pipe. We used a 15' snake and were happy with the results. Last evening, I was down in the basement and saw the basement sink clogged for no reason. We had stopped using the basement sink and wasn't down there very much. I opened the drain pipe from under the sink and inserted the snake for about 5-6 feet and removed some gunk. It still seems clogged. I am able to use the main floor kitchen but the water is moving down to the basement sink and ofcourse leaking through a joint. in the basement sink. I have a bucket in place to catch the drip. Could someone help explain how I could fix this problem.