$x^2=37\pmod {77}$ is there solution for $x$?

Question

Is there solution for $x$? $$x^2=37\pmod {77}$$

Which method should we use, Diophant equations or?

I found nothing by using induction.

thanks

Answer 1

Try reducing modulo $7$ and $11$. That should make the problem more tractable. By the Chinese remainder theorem there is a solution modulo $77$ if and only if there is a solution modulo both modulo $7$ and modulo $11$.

Answer 2

HINT:

As $77=11\cdot7,$

$$x^2\equiv37\pmod{77}\equiv37\pmod7\equiv2$$

Now, $a\equiv0,\pm1,\pm2,\pm3\pmod7\implies a^2\equiv0,1,4,9\equiv2\pmod7$

and $$x^2\equiv37\pmod{77}\equiv37\pmod{11}\equiv4=2^2$$

Use Solutions of the congruence $x^2 \equiv 1 \pmod{m}$ and CRT