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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 \(13\).

 Nov 10, 2020
 #1
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By computer program:

 

for [a = 1, a <= 12, a = a + 1]

for [b = 1, b <= 12, b = b + 1]

for [c = 1, c <= 12, c = c + 1]

  if (2*a*b + b*c + c*a % 13 == 0)

  if (a*b + 2*bb*c + c*a % 13 == 6*a*b*c % 13)

  if (a*b + b*c + 2*c*a % 13 == 8*a*b*c)

  output(a,b,c)

 

output: a = 7, b = 11, c = 3, so the answer is 8.

 Nov 10, 2020
 #2
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See numerous 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

 Nov 10, 2020
 #3
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Thanks! Both of your answers are great!

 Nov 10, 2020

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