How does gamma emission affect the atomic number of an isotope?
It doesn't change the atomic number.
The atomic number is the number of protons in the nucleus of an atom. An atom can be radioactive when the ratio of neutrons and protons is not optimal. It then decays emitting particles.
It is also possible that an atom is in a metastable state, meaning that the nucleus of the atom contains excess energy. In this case the neutron/proton ratio is ok, but the nucleus needs to lose its excess energy. The excess energy is emitted as gamma rays.
The general form of the equation for this decay is:
You can see that the atomic number, mass number and therefore the name of the isotope all stay the same!