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# If are positive integers less than 13 such that

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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 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

Dec 1, 2022

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I have seen this solved. Search for it by putting it in the "Search Box" at the top right-hand corner.

Dec 1, 2022
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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)

Dec 1, 2022