Solve simultaneous systems of congruences $x\equiv 10 \pmod{60}$ and $x\equiv 80 \pmod{350}$

Question

Solve simultaneous systems of congruences $x\equiv 10 \pmod{60}$ and $x\equiv 80 \pmod{350}$

How does one solve this using CRT? Because it has duplicate primes in its factorization? I got

$$350=5*5*2*7$$
$$60=2*2*3*5$$

So I tried CRT with

$x\equiv 1\pmod3, x\equiv0\pmod5, x\equiv0 \pmod2, x\equiv3 \pmod7$ and got the wrong answer.

or should it be
$x\equiv 1\pmod3, x\equiv5\pmod{25}, x\equiv2\pmod4, x\equiv3\pmod7$

Answer 1

Your work will be greatly simplify using the formula for the solutions of a system of two simultaneous congruences with coprime moduli $a$ and $b$ based on Bézout's identity $$ua+vb=1,\qquad(u,v\in\mathbf Z):$$ $$\begin{cases} x\equiv \alpha\mod a\\x\equiv\beta\mod b \end{cases}\iff x\equiv\beta\,ua+\alpha\,vb\mod ab.$$ Now your congruences are equivalent to $\begin{cases}x'\equiv 1\mod 6\\x'\equiv8\mod 35 \end{cases}$ and $x\equiv 10x'\mod60\cdot 350$.

Added:

The given congruences mean there exist $k,l \in\mathbf Z$ such that $x=10+60k=80+350l$, i.e. $x=10(1+6k)=10(8+35l)$, so $1+6k=8+35l$, which I denote $x'$, whence the simplified system of congruences and the relation between $x$ and $x'$. congruences

Answer 2

When the moduli are not coprime there may not be a solution. With a small system, it's probably better to solve in the following fashion so that you can see when and why a contradiction arises:

From the first congruence you have $x=10+60y$. Plug that into the second congruence to get

$$10+60y \equiv 80 \pmod{350}$$

$$60y \equiv 70 \pmod{350}$$

$$6y \equiv 7 \pmod{35}$$

$6$ is its own inverse mod $35,$ so we have

$$y \equiv 42 \equiv 7 \pmod{35}$$ or $y=42+35k$.

Then $x = 10 + 60 (42 +35k) = 2530+2100 k$, or if you shift $k$ by one, $x = 430 +2100k$.

Answer 3

It should be $\color{magenta}{x\equiv 1\pmod3}, \color{blue}{x\equiv5\pmod{25}}, \color{magenta}{x\equiv2\pmod4,} \color{brown}{x\equiv3\pmod7}$.

From $\color{magenta}{x\equiv-2\pmod{3,4}} $ we get $x\equiv-2\equiv10\mod12.$

From $x\equiv10\pmod{12}$ and $\color{blue}{x\equiv5\pmod{25}},$ we get $x\equiv130\pmod{300}$.

From $x\equiv130\pmod{300}$ and $\color{brown}{x\equiv3\pmod7}$, we get $x\equiv430\pmod{2100}$.