Question

Why does modern atomic theory say that atom is divisible?

Answer 1

The discovery of subatomic particles such as electron, proton, and neutron challenged the Dalton theory which considered atom to be indivisible.
Discovery of alpha, beta and gamma rays further aggravated the problem. Later, Rutherford's gold foil experiment established certain conclusions which indicated that the atom itself has a defined structure made by subatomic particles. Hence, considering atom to be indivisible was no more acceptable.