logitech keyboard and mouse connect to wireless

If you have an older keyboard & a newer mouse, even though they are supposedly 'the same thing' they may be using different versions of the dongle technology. Logi change the dongle requirements over time. They're either not-really-wifi or not-really-bluetooth, they won't link to anything except the 'family' they're designed for. Logitech's receivers are not interchangeable. Is there any way to pair the old keyboard and the new mouse with the same non-unifying receiver? What other alternatives do I have besides plugging to receivers simultaneously? But why can't I pair it with the older receiver (with or without the keyboard)? I also tried pairing the keyboard with the new mouse's receiver, but failed (the Connect Utility doesn't recognize that receiver at all - it says no receiver is connected - although the mouse is working just fine with it). The new mouse works with its dedicated receiver that came with it, so I know that the mouse is OK. I do as the instructions say (turn it off then on again) but the utility doesn't recognize it (I tried this strategy to re-pair the older keyboard and it worked). Now one button in the mouse stopped working so I bought a new M185 mouse, but I can't pair it with the older receiver using Logitech's Connect Utility (version 2.30.90).

I have a Logitech keyboard and a mouse pair (keyboard K270 and mouse M185) that are paired to a single non-unifying receiver, and they work (I bought them together as a pair).

Logitech keyboard and mouse connect to wireless