Is there a way of calculating the value on the opposite side of a set in modular arithematic?

Question

So mod 12 is like a clock- if the hand is pointing at 12 is there a way of calculating the number on the opposite side of the clock? ( ie. 6 ) For mod 12 it's easy but for a clock with thousands of units is there a way of calculating that value?

Answer 1

For the most part yes. Note that a clock with an odd number of sides has two different things on "the opposite side". Besides this small hiccup, we have that for an $n$-sided clock, then: $$\text{opposite}_n(a) = a + (n/2)\mod n$$ We can essentially just "wrap our number around the clock halfway", and it will work.