Bluetooth Keyboard Paring Problem

I'm not being able to connect by bluetooth keyboard to my MacBook Air.

I tried removing the batteries and using new ones.

Tried turning it off and on again with the on/off button pressed until the passcode is required.

And I tried reseting the SMC and PRAM.

When the keyboard appears on the list of devices in Bluetooth Preferences and click to pair it, and it shows the passcode.

I press the numbers in the keyboard and 'return' and it says every time: the passcode is incorrect.

So I tried pressing the passcode from the built in keyboard, it does not says is incorrect, but doesn't pair anyway.

I searched and didn't find any more suggestions to what I could do.

Can anyone help me?

The keyboard seems to be working perfectly, the light turns on, show the searching mode, and turns off the way it is supposed to....

It has been used with 2 ipads, but they're not paired anymore, and not in 1km range from the keyboard now.

Also, I tried pairing it with my iPhone 6. It also says the 4 digit passcode is incorrect and that the "connection failed".

This is my last hope, I live in Brazil, and there's no Apple Store close to me.

Thanks in advance!