# A problem on chinese remainder theorem (CSIR NET DEC 2015)

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.

It is only unique modulo $5\cdot 7\cdot 11$.
In particular, $752$ and $1137$ are solutions.
• @FairoosaNavas it sounds like you need to look at the Chinese remainder theorem again,for example sample at Wikipedia. It clearly says that the solutions are equivalent mod $385$ in your case, which is how I've given you these two other solutions – rschwieb Mar 4 '16 at 4:07