+177 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 \(13\).
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.
