I saw someone ask a question regarding the congruences below. $c≡1 \mod 143$
$c≡315 \mod 323$
$c≡167 \mod 667$
I tried to solve them with this equation: For any number of congruences $x≡a\mod n$, $N$ is the product of the moduli.
$c= \left[\Sigma_i = a_i \frac N{n_i} \left[\left(\frac N{n_i}\right)^{-1}\right]_{n_i}\right]_N$
I won't post all my the Extended Euclidean algorithm stuff but essentially these are the results I got:
$1\left( \frac {30808063} {143} \right)\left[\left( \frac {30808063} {143} \right)^{-1}\right]_{143}$ +
$315\left( \frac {30808063} {323} \right)\left[\left( \frac {30808063} {323} \right)^{-1}\right]_{323}$ +
$167\left( \frac {30808063} {667} \right)\left[\left( \frac {30808063} {667} \right)^{-1}\right]_{667}$
$ = (215441*122 + 315*95381*286 + 167*49189*221 )\mod 30808063 = 3154410$
Is this correct?