The pic related shows a chart of a series of the products of Arabic numerals multiplied by the digts 1-10 in order, as well as the numerological value of each product, where you add the total amount of digits until you arrive at a single number (e.g., 80 = 8+0 = 8; 72 = 7+2 + 9; and 56 = 6+5 = 11 = 1+1 = 2; etc.)

The highlighted values demonstrate that 9 and 0 are the only ones that produce the same result regardless of the number by which you are multiplying them (and multiplication is just serial addition), and thus the only digits that can be ignored in terms of affecting the numerological value.

This has been noted in the past as the practice of "casting out the nines," wherein you can simply ignore the values that add to 9, because they will have no effect on the outcome, just as you can ignore zeroes in terms of their effect on a sum, numerologically speaking.

But there appears to be a pattern that results when you perform such a calculation that literally treats both 9 and 0 as numerically equivalent, and I'm honestly wondering how to mathematically explain it, because it troubles me.

I don't know how obvious this pattern is, but I will attempt to explain it.