If you know only the atomic number, you can estimate the mass number of the most common isotope of the element by looking up the average atomic weight of the element and rounding it to the nearest integer. For example, the number of neutrons in the most common isotope of lithium (atomic weight 6.941 amu and atomic number 3) is 7-3 = 4.
Experimentally, the mass numbers are determined by mass spectrometry. The element is ionized, accelerated in an electric field, and then passed through a strong magnetic field. The field deflects light ions more than heavy ones with the same charge, and (with proper calibration) the amount of deflection can be related to the ion's mass number.
The atomic number can be determined by observing the element's x-ray spectrum. The frequencies of x-ray radiation emitted when the element is used as a target in an x-ray tube have a simple relationship with the element's atomic number.
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