A phosphorus atom needs to gain ---electrons to achieve a full octet

Three (3).

Explanation:

The octet rule is a rule in chemistry that states that, chemical elements would either gain or loss electrons in their outermost shell in order to attain a stable electronic configuration i.e having eight (8) valence electrons in their outermost shell such as chemical elements found in group 8 (2-8-8) known as noble gas.

A phosphorus atom needs to gain three (3) electrons to achieve a full octet because the atomic number of phosphorus is 15 and as such it has the electronic configuration of 2-8-5 and needs three electron in order to become a stable element.