Three diophantine equations

Question

I have three Diophantine equations that I need to solve. However, I am struggling to find the values of $x$, $y$, and $z$. To clarify, all equations equal the same thing.

The approach that I took was to equate a) and b), solve for $x$ and $y$, and then get the value for $z$. Is this approach correct or is there a different way to go about solving it? How would I solve the below equations?

a) $c = 143x + 1$

b) $c = 323y + 315$

c) $c = 667z + 167$

Answer 1

If $x$, $y$, and $z$ have to be integers, then $$ \begin{align*} x &= \frac{c - 1}{143} \\ y &= \frac{c - 315}{323} \\ z &= \frac{c - 167}{667} \end{align*} $$ so $$ \begin{align*} c &\equiv 1 \pmod{143} \\ c &\equiv 315 \pmod{323} \\ c &\equiv 167 \pmod{667} \end{align*} $$

Now we can use the Chinese Remainder Theorem as 143, 323, and 667 are pair-wise relatively prime. Writing $M = 143 \cdot 323 \cdot 667$, we have

1. $b_1 \cdot M / 143 \equiv 1 \pmod{143} \implies b_1 \equiv 112$
2. $b_2 \cdot M / 323 \equiv 1 \pmod{323} \implies b_2 \equiv 286$
3. $b_3 \cdot M / 667 \equiv 1 \pmod{667} \implies b_3 \equiv 221$

Thus $c \equiv 1 \cdot 112 \cdot M / 143 + 315 \cdot 286 \cdot M / 323 + 167 \cdot 221 \cdot M / 667 \equiv 1000000 \pmod{M}$.

All solutions are therefore given by $c = Mk + 1000000$, where $k$ is an integer, and $x$, $y$, and $z$ as described in the first three equations.