#Na^+# is a sodium atom that has lost an electron to form a positive ion. Sodium has an atomic number of 11 and so therefore in a neutral state, one atom of sodium has 11 protons and 11 electrons. If we lose an electron we then have 10 electrons.
#F^-# is a fluorine atom that has gained an electron to form a negative ion. Fluorine has an atomic number of 9 and so therefore in a neutral state, one atom of fluorine has 9 protons and 9 electrons. If we gain an electron we have 10 electrons (same as a #Na^+# ion).
To be neutral, an atom has to have an equal number of protons (+1 charge) as electrons (-1 charge) . So if our 'mystery' element has 10 electrons, it must also have 10 protons.
The number of protons in an atom is given by its Atomic Number in the periodic table. The element with an atomic number of 10 is Neon (Ne).