MICROSOFT ARC TOUCH MOUSE - a question of life and death?

Let me first finish mainstream political correctness and then as second point: reality.

1. Women are perfect, know everything, are technical genius and born for engineering.

2. I have a cleaning woman in whose head the word "wireless" falls into a large hole which I am unable to fill since two years. "Not connected by cable" for her is "not connected" meaning " to arrange" meaning "to make disappear". I have 2 desktop PCs and 5 Laptops. All with wireless keyboards and Microsoft Arc Touch Mouses. Every week my cleaning woman makes a little high pile of 7 Arc Touch mouses (the glue together being MAGNETIC) and sometimes she opens her hand and proudly presents 7 little black USB receivers (because I said I shoot her when she throws away the next one). When I tell her that in the next future a newspaper headline will say: "Cleaning woman tortured slowly to death" and when they will get me, I will shout: "I enjoyed every second of it" she doesn't even get the joke. Before I spend again hours to find the right mouse, receiver, computer and / or keyboard, my question:

The USB receiver dongle as the mouse both on the inside of the battery holder have a lot of long numbers.

For example: on the receiver 201WW10215005 ; and: FCO ID C3K1447 and: CGA169LP1380T0 Is one of those numbers one with which the mouse can be paired with the corresponding receiver and if so: which one?

(Yes, I could buy a bluetooth mouse but I LOVE the Arc Touch).