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\(If $a,b,c$ are positive integers less than $13$ such that \begin{align*} 2ab+bc+ca&\equiv 0\pmod{13}\\ ab+2bc+ca&\equiv 6abc\pmod{13}\\ ab+bc+2ca&\equiv 8abc\pmod {13} \end{align*}then determine the remainder when $a+b+c$ is divided by $13$.\)

Please help I really am bad at these sorts of things.

Thanks!

 Jan 5, 2019

Best Answer 

 #1
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Check here and see how many solutions there are:

 

https://www.wolframalpha.com/input/?i=2ab+%2B+bc+%2B+ca+%3D0,+%5Bab+%2B+2bc+%2B+ca%5D+mod+13+%3D6abc,+%5Bab+%2B+bc+%2B+2ca%5D+mod+13+%3D8abc,+solve+for+a,+b,+c

 Jan 5, 2019
 #1
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+1
Best Answer

Check here and see how many solutions there are:

 

https://www.wolframalpha.com/input/?i=2ab+%2B+bc+%2B+ca+%3D0,+%5Bab+%2B+2bc+%2B+ca%5D+mod+13+%3D6abc,+%5Bab+%2B+bc+%2B+2ca%5D+mod+13+%3D8abc,+solve+for+a,+b,+c

Guest Jan 5, 2019

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