Question: I managed to get myself in a two-factor authentication pickle. My Apple id is associated with a Mac (currently running OS X Yosemite), an iPhone and an iPad.

The iPad just came back from repair and I needed to restore it from the cloud. Unfortunately before doing that I lost my iPhone! I now have nothing that can receive the two-factor authentication codes. (Didn't see that one coming!)

My Mac is tied to my iCloud account and Apple Id, and I see that Sierra can receive these codes and that I can upgrade from Yosemite which is what's currently running on the Macbook Pro.

If I upgrade to Sierra will that mean my Mac can then also receive the two factor codes (allowing me then to get my iPad back up and running)? 

Or will I be out of luck as I may need to use two-factor to get the laptop properly set up with Sierra? If that's the case then I do not want to upgrade as it means I may lose the limited iCloud access that I have via the Mac.

Help!

N.B. Yes, have a full trustworthy backup of the Mac on Time Machine.

Answer: 
You can request the two factor code to be sent to your phone number instead. Click the "Didn't get a code" on iPad, where you're asked to enter the code. Then you can get it sent to your trusted phone number.

The trusted phone number doesn't have to be an iPhone, so you can use any dumb phone that can receive SMS text messages.
I am fairly confident that the answer to this is “No, the device is not yet trusted and updating the operating system will not change that.”

The reason being that having regained my phone and therefore the ability to get two factor codes I see my devices described as “trusted” with the exception of the laptop. 

It seems unlikely that the laptop could be considered trusted until it is explicitly signed in by one of the other trusted devices. 

If I feel brave one day I may upgrade the OS and see if that is correct. 

(Oh, and yes, lesson learned - I have added a whole bunch of extra backup numbers so I can get verification codes through multiple avenues!)

**Update**

The answer is indeed “no”. Having updated to Sierra I cannot be fully integrated with iCloud without getting it in to the trust circle.