+297 If \(a,b,c\) are positive integers less than 13 such that
\(\begin{align*} 2ab+bc+ca&\equiv 0\pmod{13}\\ ab+2bc+ca&\equiv 3abc\pmod{13}\\ ab+bc+2ca&\equiv 8abc\pmod {13} \end{align*}\)
then determine the remainder when \(a+b+c\) is divided by 13.
Answer with solution will be appreciated
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Notice that if b = 2a and c = 3a the left hand side of the first equation becomes 4a2 + 6a2 + 3a2 = 13a2
In other words it is a multiple of 13 and hence satisfies the first equation.
For this situation, if a, b and c are all positive integers less than 13, then a can only be one of 1, 2, 3 or 4.
Trying these in turn we find that only a = 3 (hence b = 6 and c = 9) satisfy the second and third equations.
Hence a + b + c = 5 mod(13)
