If $a, b, c$ are positive integers less than 13 such that
$2ab+bc+ca \equiv 0\pmod{13}$
$ab+2bc+ca \equiv 6abc\pmod{13}$
$ab+bc+2ca \equiv 8abc\pmod {13}$
then determine the remainder when $a + b + c$ is divided by $13$.
Look at many, many solutions here:
https://www.wolframalpha.com/input/?i=%282ab%2Bbc%2Bca%29+mod+13+%3D0%2C++%28ab%2B2bc%2Bca%29+mod+13%3D+6abc%2C++%28ab%2Bbc%2B2ca%29++mod+13+%3D+8abc%2C+solve+for+a%2Cb%2Cc
+118667 If a, b, and c are all 0 then these three equations are all true.
And if this is true then (a+b+c) mod13 = 0
So if there is only one solution then that solution must be 0
However, just two of them are 0 then the third one can equal to many different numbers.
So
I do not believe this has one specific answer.
I know I must be annoying for saying all of the answers are wrong, but unfortunately there's a problem with this one too. The problem states that $a, b, c$ must be positive integers less than 13, so $0$ doesn't fit this requirement. However, thank you very much for trying to solve this for me!
a = 3, b = 6, c = 9.
Unique.
Will post a solution, of sorts, if anyone is interested.
+118667 Thanks guest,
I have some interest.
Did you use code to do the grunt work and just test all the possibilities
Or did you actually come by the answer in a logical, mathematical (no trial and error) way ?
Thanks Melody, it's nice to know that someone is actually interested.
Often that doesn't appear to be the case.
In answer to your question, I arrived at the solution algebraically.
The equations can be written in the form
\(\displaystyle 2X+Y+Z\equiv 0 \dots\dots(1)\\ X+2Y+Z\equiv R\dots\dots(2)\\ X+Y+2Z\equiv S\dots\dots(3)\)
where each one is mod 13.
\(\displaystyle (1) - 2(3) : -Y-3Z \equiv -2S\dots\dots(4)\\ (2) - (3) :Y - Z \equiv R-S\dots\dots(5)\\ (4)+(5) : -4Z\equiv R-3S\dots\dots(6).\)
Returning to the original unknowns,
\(\displaystyle -4ca \equiv 6abc-24abc \equiv -18abc,\)
from which
\(\displaystyle9b=2+13k\quad \text{ where k is an integer}.\)
The smallest value of k for which b will be an integer is k = 4, leading to 9b = 54, b = 6.
The next suitable value for k puts b out of the permitted range.
The remaining unknowns (a = 3 and c = 9) can then be found by back substitution, (6) into (5) and then into (2) or (3).
Leave that to you.
