Solving of system of congruent residues

Question

let us suppose we have following two equation

$\rm\ y+8\equiv 4\pmod {11} $

$\rm\ y+8\equiv 7\pmod {11}$

for example for the first equation , we can get following values of $y$

$y=7, 18, 29$ for instance

for the second equation we can get following solutions

$y=10, 21, 32$

but on one site , there is given solutions like this

$\rm\ y\equiv 7\pmod {11} $

$\rm\ y\equiv 10\pmod {11} $

How can I find it? Using Chinese remainder theorem or?

add 3 to each side of the equation... and you'll discover that substracting X is the same than adding 11-X. — zwim, Mar 04 '17 at 08:34
@dato datuashvili: is it possible that you have made a typo in your problem statement: we have that $4 \equiv y + 8 \equiv 7 \mod 11$ but $4$ is not congruent with $7$ $\mod 11$. — Student, Mar 04 '17 at 08:55
@zwim: sorry, i was confused because of the title stating it is a system of equations. — Student, Mar 04 '17 at 09:34

score 0 · Answer 1 · answered Mar 04 '17 at 08:57

0

If the two equations are supposed to be a system of equations then there is no solutions.

If those two are separate equations just add $3$ to both sides to get the answer.

answered Mar 04 '17 at 08:57

Harto Saarinen

can you tell me general rule about this? – dato datuashvili Mar 04 '17 at 08:58
Well in the first case there is no solution because a number cannot be congruent to two different numbers when the modulus is same. And in the second it's just the fact that $11=0$ (mod $11$). – Harto Saarinen Mar 04 '17 at 09:02
does this rule work for any similar case? – dato datuashvili Mar 04 '17 at 09:13
i will post one more question – dato datuashvili Mar 04 '17 at 09:33

1 Answers1