The pattern you described in the periodic table, where the number of valence electrons in groups 1-2 and 13-18 corresponds to the ones digit of the group number, is not a mere coincidence. It arises from the electronic configuration and organization of elements.
The periodic table is structured based on the electron configuration of atoms. The elements in groups 1-2 and 13-18 are known as the "s-block" and "p-block" elements, respectively. The group number represents the number of valence electrons in these elements. Valence electrons are the electrons in the outermost energy level (or shell) of an atom, which are primarily responsible for an element's chemical properties and reactions.
In the s-block elements (groups 1-2), the valence electrons are located in the s subshell. Each s subshell can hold a maximum of 2 electrons. Therefore, group 1 (e.g., hydrogen, lithium, sodium) has 1 valence electron, and group 2 (e.g., beryllium, magnesium, calcium) has 2 valence electrons.
In the p-block elements (groups 13-18), the valence electrons are located in the p subshell. Each p subshell can also hold a maximum of 2 electrons per orbital (px, py, pz), resulting in a total of 6 valence electrons in this block. The group number in the p-block corresponds to the total number of valence electrons, starting from 3 (group 13) and ending with 8 (group 18).
If we were to use a different base number system other than the decimal (base 10) system, the pattern might still exist, but the specific digits representing the number of valence electrons in each group would change. For example, if we used a base-8 system (octal), the pattern would persist, but the valence electrons in each group would correspond to the ones digit in the octal representation of the group number. Similarly, in a base-16 system (hexadecimal), the valence electrons would correspond to the ones digit in the hexadecimal representation of the group number.
In summary, the relationship between the group number and the number of valence electrons in groups 1-2 and 13-18 of the periodic table is not coincidental but rather a consequence of the electron configurations and the organization of elements.