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# hard hard hard

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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