If are positive integers less than 13 such that

Question

-1

207

2

+307

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

Imcool Dec 1, 2022

Guest · Answer 1 · 2022-12-01T12:38:11

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

Guest · Answer 2 · 2022-12-01T12:46:14

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)