Why would an element be radioactive?
For an element to be stable, its neutron to proton ratio should be within the stability range. Such stability range graphs have been found and plotted with neutron/proton ratio vs. number of protons. If for any element, corresponding to the number of protons in it, the ratio of neutron to proton does not lie within the stability range, then that element will not be stable and will be radioactive. It will break up into smaller elements giving out α and β particles as may be necessary to bring the neutron to proton ratio within the stability range and this process of breaking up into smaller pieces continues till the final products are non-radioactive.