Which of the following intervals contains an integer satisfying following three congruences
$$x=2\pmod5\\
x=3\pmod7\\
x=4\pmod{11}$$
$a) [401,600] \\ b)[601, 800] \\ c)[801,1000] \\ d)[1001,1200]$
(CSIR NET 2015 Dec)
I tried this question and I got answer but it is not in the option.
I applied Chinese remainder theorem.
$$x=2\pmod5\\
x=3\pmod7\\
x=4\pmod {11}$$
$$N_1=7\times11=77\\
N_2=5\times11=55\\
N_3=7\times5=35$$
$77x=1\pmod5\implies b_1=3\\
55x=1\pmod7\implies b_2=6\\
35x=1\pmod {11} \implies b_3=6$
then,
$x=2\times77\times3+3\times55\times6+6\times35\times4=2292$
This answer is not in the option.
If my work is wrong please correct it.
Best Answer
You know that the answer you get applying the CRT is not a unique integer, right?
It is only unique modulo $5\cdot 7\cdot 11$.
In particular, $752$ and $1137$ are solutions.
I'll leave you with the tasks of deciding if this is complete, and how you will use it to answer your question.