TITLE: Justifying a cryptic question: find $x$ if $x-5=11$ and $x+5=9$
QUESTION [6 upvotes]: A cryptic question being asked on the Social Media:
$$\text{Find}~ X,~~
 \text{if}~ X-5=11, ~\text{and}~ X+5=9.~~~~(1)$$
is mathematically vague. But an interesting answer given is 16 and it refers to a clock with two hands when it shows 4 O'clock.
I find that using mod(modular)-arithmetic we can justify it.
By M(mod N) we mean the remainder when the  M is divided by N.
For instance: 17(mod 15)=2, 12(mod 15)=-3 or 12 so on and so forth.
This way the cryptic equation (1) become a sensible mod-equation as
X(mod 12)-5(mod 12)=11(mod 12)
and
X(mod 12)+5(mod 12)=9(mod 12)
$\implies$ X=16.
The question is: Am I right?

REPLY [11 votes]: In any number system (ring) the following equivalences hold true  $$\begin{align} &x\!-\!5=11,\ x\!+\!5= 9\\[.3em]
\iff\  \, &\ 11\!+\!5\,=\, x\, =\, 9\!-\!5\\[.4em] 
\smash{\overset{{\rm subtract}\ 4}\iff}\ \  &\ \ \ \ \ \color{#c00}{12} = x\!-\!4 = \color{#c00}0\end{align}\qquad$$
So your equations have a solution in a ring $\iff \color{#c00}{12 = 0}\,$ in the ring. For example, they are solvable in $\,\Bbb Z_n =\,$  ring of integers $\bmod n\,$ where $\,n\,$ divides $\, 12,\,$ i.e. $\, n = 2,3,4,6,12,\,$ as well as any rings containing such $\,\Bbb Z_n$ (i.e. $\Bbb Z_n$-algebras) e.g. rings of polynomials, power series and matrices over $\,\Bbb Z_n$. 
In any such ring the equations have unique solution $\,x = 4.\, $ Note that your solution is identical because  $16 = \color{#c00}{12}+4 = \color{#c00}0+4 = 4\,$ by $\,\color{#c00}{12 = 0}\,$ in every such ring.