Solving Quadratic Congruences: 16 Solutions

[talolard]

Homework Statement

How many solutions does $$x^{2}\equiv9 mod 7700$$ have?
So my question is if this solution is "legitimate"
Solution

First notice that$$7700=7\cdot11\cdot2^{2}\cdot5^{2}$$

Thus we must solve the system $$\begin{cases} x^{2}\equiv2 & \left(7\right)\\ x^{2}\equiv9 & \left(11\right)\\ x^{2}\equiv\mbox{1} & \left(4\right)\\ x^{2}\equiv9 & \left(25\right)\end{cases}.$$
This system is equivelent to
$$\begin{cases} x\equiv\pm2 & \left(7\right)\\ x\equiv\pm9 & \left(11\right)\\ x\equiv\pm1 & \left(4\right)\\ x\equiv\pm9 & \left(25\right)\end{cases}.$$
since gcd(1,p)=1 is always true, all of these equations are solvable by the fundamental theorem of modulo arithmetic. There are [tex]2^{4}=16[tex] disjoint equations and thus 16 distinct solutions.

Each solution must be distinct, since if x solves to different equations, then for instance we would have [tex]x\equiv2\equiv-2\left(7\right)[tex]

Homework Equations

The Attempt at a Solution

 

[mighty2000]

it is important to approach problems with a critical and analytical mindset. While the solution provided may seem valid at first glance, it is important to thoroughly examine the steps and assumptions made.

Firstly, it is unclear what the goal of solving this system of equations is. The original question asks for the number of solutions for x^{2}\equiv9 mod 7700, but the solution provided solves for x in four different congruences. It is important to clarify the objective before proceeding with a solution.

Secondly, the solution assumes that all four congruences are solvable by the fundamental theorem of modulo arithmetic. While this may be true in most cases, it is not always the case. It is possible for some of the congruences to have no solutions or for the solutions to be non-distinct.

Lastly, the solution jumps to the conclusion that there are 16 distinct solutions without providing any explanation or justification. It is important to show the steps taken to arrive at this conclusion and to provide a clear explanation for why there are 16 solutions.

In summary, while the solution provided may seem reasonable, it is important to approach problems with a critical and analytical mindset and to thoroughly explain and justify each step taken.