Working out the number of solutions......

Question

How would I go about figuring out the number of solutions in $x^2=-1211 ($mod $2020)$. I wrote out the prime factorisation of $2020$ which is $2^2 \times 5 \times 101 $but I sort of found myself getting very confused afterwards, watching several videos seemed to make that even worse so some step-by-step explanations would really help!

Answer 1

hint

Your equation is equivalent to$$x^2\equiv 809 \mod 2020$$

Answer 2

Hint:

Use the Chinese remainder theorem. Note that $$1211\equiv \begin{cases}3\mod 4,\\1\mod 5, \\ 100 \mod 101.\end{cases}$$

Answer 3

Something no one else has mentioned...

Let S = {0, 1, 2, ..., 100}.

One of the steps that you will need to take is to identify all elements x from set S such that $x^2 \equiv -1211 \pmod{101}.$

I know of no straight forward way of mathematically identifying the relevant elements from S. If I were attacking this problem, I would write a computer program to identify the corresponding elements from S. Then, as has already been suggested, I would educate myself re the Chinese Remainder Theorem.