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\(If $a,b,c$ are positive integers less than $13$ such that\begin{align*} 2ab+bc+ca&\equiv 0\pmod{13}\\ ab+2bc+ca&\equiv 6abc\pmod{13}\\ ab+bc\)

 

yes, I have tried lots of stuff, adding them together but that doesn't work, I used wolfram and it doesn't even give me an answer

halpppp

 Oct 15, 2020
 #1
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See some 50+ solutions here:

 

https://www.wolframalpha.com/input/?i=%282*a*b%2Bb*c%2Bc*a%29+mod13%3D%3D0%2C+%28a*b%2B2*b*c%2Bc*a%29mod13%3D%3D6*a*b*c%2C+%28a*b%2Bb*c%2B2*c*a%29mod13%3D%3D8*a*b*c%2C+integer+solution

 
 Oct 15, 2020
 #2
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Using Wolfram Alpha, I get a + b + c == 2 + 8 + 1 == 11 (mod 13).

 
 Oct 15, 2020

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