An atomic number is the number of protons in the nucleus and that is what determines the identity of an element. The mass number is the total of protons and neutrons in the nucleus. An element has a distinctive Atomic number, but many elements can have the same mass number. For example Carbon-12 has an atomic number of 6 and a mass number of 12. Nitrogen-12 has an atomic number of 7 but the mass number is 12. Oxygen-12 has an atomic number of 8 but the mass number is still 12.
By the way, the atomic number initially had nothing to do with the protons in the nucleus, but came about from a study of how elements scattered X-rays. It is the order that was found in this study.
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