From a GRE book: "The units digit of a product of positive integers is equal to the units digit of the product of the units digits of those integers."
I read this and was thinking... why would you not write this symbolically?
Here is my attempt:
Let $x_1, \dots, x_n$ be in $\mathbb{Z}_{\geq 0}$ and let $\mathcal{U}: \mathbb{Z}_{\geq 0}\to \mathbb{Z}_{\geq 0}$ be defined by $$\mathcal{U}(x) = \text{units digit of }x\text{.}$$ Then $\mathcal{U}\left(\prod\limits_{i=1}^{n}x_i\right) = \mathcal{U}\left(\prod\limits_{i=1}^{n}\mathcal{U}(x_i)\right)$.
If you know of a simpler way to write this than what I have above, that would be appreciated as well.
For some set of positive integers $A$: $$ \Big(\prod_{n \in A} (n \!\!\!\!\mod 10)\Big)\!\!\!\!\mod 10 = \Big(\prod_{n \in A} n\Big)\!\!\!\!\mod 10 $$