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The periodic table of (known) elements is organised in order of atomic number (from lowest in the top left hand corner, to highest in the bottom right corner). The atomic number is the number of protons inside the 'atomic nucleus'. In a stable atom, the negatively charged electrons equal the number of positively charged protons. Since the periodic table only lists neutral elements we therefore know that the number of electrons is the same as the number of protons. Finally, to find out the number of neutrons, we can subtract the atomic weight from the atomic number (giving us what is 'left over').
If we use Oxygen as an example, it has an atomic number of 8, so we know it has 8 protons and 8 electrons. To find the neutrons, we can take 8 away from 16 (the average atomic weight 15.999 rounded up). So conveniently, Oxygen has 8 protons, 8 electrons and 8 neutrons!
To conclude:
# of Protons = Atomic number
# of Electrons = # of Protons.
# of Neutrons = Atomic weight minus Atomic number.
Have a go at building an atom using your knowledge of the periodic table.