Modular Number Theory

We know that the integers modulo  13  form a field, so we can operate on systems almost like usual. Equalities will just mean congruence modulo  13 .
From the first equation we obtain  ca=−2ab−bc  and therefore the second equation gives
ab+2bc−2ab−bc=2b(−2ab−bc) 
that is
b(a−c)=2b2(2a+c) 
The third equation gives  
 ab+bc−4ab−2bc=8b(−2ab−bc) 
that is:
b(3a+c)=8b^2(2a+c) 

We know that  b≠0  (because  b  is a positive integer less than  13 ), so we can simplify it off. If  2a+c=0 , then also  a−c=0 , but this implies  a=c=0 , which is excluded. Thus we obtain
2b=(a−c) / (2a+c) 
8b=(3a+c) / (2a+c) 
Thus we need
4(a−c) / (2a+c)=(c+3a) / (2a+c) 
and therefore  (4a−4c)=(c+3a) , that is,  a=5c .
 Now we can substitute in  2ab+bc+ca=0  to get
10bc+bc+5c^2=0 
Since  c≠0  we obtain  11b+5c=0 , so  2b=5c  Multiplying by  7  yields  14b=35c  and so  b=9c
 Now we can substitute in
ab+2bc+ca=2abc 
to get:
45c^2+18c^2+5c^2=90c^3 
Thus:
  3c^2=12c^3 
and so  −c=3 , that is,  c=10 . Hence  a=5c=11  and  b=9c=12 
Solution:
a=11,b=12,c=10 
and so  a+b+c≡7(mod13)         

I'd like to see an answer for this too.    

Solution:

$\text{Modular arithmetic theorem: }\\ \text {If (a, b, c) are positive integers and} \pmod{m} \text { is relatively prime, then there exists }\\ (a^{-1}, b^{-1}, c^{-1}) \text { such that } \left(a*a^{-1} \equiv b*b^{-1} \equiv c*c^{-1} \pmod{m} \right), \\\text{ with a, b, c, relatively prime within the ring of modulo (m); i.e. } 1\leq [a’, b’, c’] < 13.$

$\begin{array}{|rccc|} \hline a^{-1}b^{-1}c^{-1} (2ab + bc + ca) \equiv 1 \pmod{13}\\ a^{-1}b^{-1}c^{-1} (ab + 2bc + ca) \equiv 3 \pmod{13} \\ a^{-1}b^{-1}c^{-1} (ab + bc + 2ca) \equiv 5 \pmod{13}\\ \hline\end{array} \Rightarrow \begin{array}{|rccc|} \hline 2c^{-1} + a^{-1} + b^{-1} \equiv 1 \pmod{13}\\ c^{-1} + 2a^{-1} + b^{-1} \equiv 3 \pmod{13}\\ c^{-1} + a^{-1} + 2b^{-1} \equiv 5 \pmod{13}\\ \hline\end{array}$

$\text {add equations: }\\ 4c^{-1} + 4a^{-1} + 4b^{-1} \equiv 9 \pmod{13}\\ 4(c^{-1} + a^{-1} + b^{-1}) \equiv 9 \pmod{13}\\ $

$\text {Note that} -4\pmod{13}\equiv 9, \text{ so } \left (a^{-1} + b^{-1} + c^{-1}\right) \equiv -1\\ \text {and } -1 \equiv 12 \pmod{13},\text { so search for three numbers where } 1 \leq a,b,c < 12.\\ \text {using the above inverse equations to find which value belongs to each variable. }\\ \text{Then take the Modular Multiplicative Inverse of these numbers, sum them to (S) and find } \mathrm S\pmod{13}. \\ \text {if you do this correctly, you will find the answer (remainder) is } \boxed {2}\\ $

GA

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Guest’s solution is correct. Mine is NOT.

My partial solution (copied from my archive of solved equations) is based on similar equations with a different set of congruencies (mod 13).

$\begin{align*} 2ab+bc+ca&\equiv abc \pmod{13}\\ ab+2bc+ca&\equiv 3abc\pmod{13}\\ ab+bc+2ca&\equiv 5abc\pmod {13} \end{align*}$

I failed to modify the congruencies to match the question.  

However, this solution method will work for the equations in the OP’s question.

GA

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Thanks Guest and Ginger   

Thanks, these were really helpful