Final answer:
The distinctive properties of an element are determined by its number of protons, which is known as the atomic number and defines the element's identity.
Step-by-step explanation:
The characteristic that gives an element its distinctive properties is its number of protons. The number of protons in an atom's nucleus is known as its atomic number (Z), which is the defining trait of an element. This atomic number determines the identity of the element, as no two different elements can have the same number of protons in their nuclei. For instance, carbon atoms always have six protons, which defines them as the element carbon, with the atomic number 6. It is important to note that while isotopes of an element may have differing numbers of neutrons, they still have the same number of protons and therefore display the chemical properties.