what is the atomic mass of the most common isotope of silver as indicated by the periodic table? why isnt the atomic weight listed on the periodic table an integer?
The periodic table is not useful for predicting the most stable isotope of any element. You can make a decent guess, but it's likely you'd be wrong much of the time. The periodic table was not constructed for nuclear chemistry; it's best to look at the chart of the nuclides (aka, table of isotopes) to answer this question.
The masses on the periodic table are weighted averages for all of the stable isotopes of each element. (For the elements that have no stable isotopes, it's the most common radioactive isotope.) It may seem rather strange that the atomic masses of each stable isotope are not integers either. Only carbon-12 has an integer for its atomic mass; it's defined that way. The other elements are all compared to carbon-12. The reason for this has to do with the non-integer masses of protons and neutrons, and the mass lost when the protons and neutrons bind together to form the nucleus. If you study nuclear chemistry long enough, you'll see that Einstein's famous equation, E = mc2, applies here.
The periodic table would predict 108 for the most stable silver isotope, but there are only two stable isotopes of silver and 108 isn't one of them. Turns out the silver-107 and silver-109 are almost equally abundant, making the average atomic mass nearly 108. Once you know that the two isotopes have masses of nearly 107 and 109, you can determine which one is more abundant using the 107.9 g/mol shown on the periodic table. Because 107.9 is closer to 107 than it is to 109, then you know that silver-107 is slightly more abundant. I hope this helps a bit. I like the other two answers too.