A, B and C are trying to think of a code. A remembers that after dividing by 13, the residue is 8. its double increased by 1234 after dividing by 17 leaves us residue of 7. C remembers that its triple increased by 4321 after dividing by 19 gives residue of 6.
I've put together the following congruences:
x = 8 (mod 13)
2x + 1234 = 7 (mod 17)
3x + 4321 = 6 (mod 19).
Using the formulas: $x = x_0 + k(\dfrac{m}{gcd(a, m)})$, whereas $ax_0 + my_0 = b$. I've come to this:
x = 8 mod 13
x = 10 mod 17
x = 7 mod 19.
After that I've used Chinese Remainder theorem.
M1 = 323; x1 = 6; M2 = 247; x2 = 2; M3=221; x3 = 8.
I get 32820 mod 4199 = 3427 + 4199*k
The result is supposed to be 6586, but I fail to produce it. Can someone please tell me where the mistake is?