The question I have is styled in the following:
"Consider the natural isomorphism of commutative rings:"
$\varphi:\mathbb{Z}/(1088) \rightarrow \mathbb{Z}/(17) \oplus \mathbb{Z}/(64)$
Write down a formula for the inverse isomorphism $\varphi^{-1}$ and calculate $\varphi^{-1}(\bar{1},\bar{17}$).
The only example I have, which I am also struggling to understand, is:
$\varphi:\mathbb{Z}/(77) \rightarrow \mathbb{Z}/(7) \oplus \mathbb{Z}/(11)$
$31\rightarrow(3,9), 1 = (-3)7 + 2(11)$
$(b,c)\rightarrow -21c+22b \mod77$
$(5,8)\rightarrow -168+110 \mod77 = 19 \mod77$
Is there a specific reason that 31 was chosen? I assume it has something to do with 31 and 77 being coprime if it was chosen for a certain reason. When I asked the professor for my course he didn't really give me a good explanation. If I could understand why the example is set out I should be able to apply it to the original question. I don't have issues with finding the inverse for a group, but for a ring I find it to be a bit more confusing.