Solving the following system of equations

Question

Okay, so I have to solve the following system of equations.

$$a\equiv 0\hspace{5mm} (\text{mod} \mathbb{Z}/5\mathbb{Z})$$ $$3-a^2\equiv 0\hspace{5mm} (\text{mod} \mathbb{Z}/5\mathbb{Z})$$

For the first equation we have that $a=5n$ for some $n\in\mathbb{Z}$. Substituting this expressions of the second equation we get:

$$3-25n^2=5m$$ for some $m\in\mathbb{Z}$. Rearranging we get $3=5(m+n^2)$ which never verifies. This means that there is no solution to this system. Is my reasoning correct?

$a\equiv 0$ implies $3\equiv 0\bmod 5$ in the second congruence, so $5$ divides $3$, which is a contradiction. — Dietrich Burde, Oct 12 '23 at 10:38

score 0 · Answer 1 · answered Oct 12 '23 at 10:38

0

The rearrangement should give you $3 = 5(m + 5n^2)$, but yes your reasoning is correct.

answered Oct 12 '23 at 10:38

Kovomaka

713

Solving the following system of equations

1 Answers1