Fix three distinct primes p, q, r, prove that the map
$Z_{pqr} → Z_{pq} × Z_{qr} × Z_{pr}$ by $[x]_{pqr}$ → ($[x]_{pq}$, $[x]_{qr}$, $[x]_{pr}$)
is injective and determine its image.
My attempt:
To prove it is injective, I said let there be $x$ and $y$ such that $[x]_{pq}=[y]_{pq}$. This means $pq|(x-y)$ So, one of $p$ or $q$ must divide $(x-y)$. Similarly, either $p$ or $r$ and either $q$ or $r$ must divide $x-y$ as well. Therefore, one of p,q,r must divide $x-y$.
This implies $pqr|(x-y)$, So, $[x]_{pqr}=[y]_{pqr}$ This proves that the function is injective (if I'm correct in my implications).
Now, how do I determine its image?
$pq \mid x-y$, so we have $p \mid x-y$ AND $q\mid x-y$
$qr \mid x-y$, so ALSO $r\mid x-y$
As $p$, $q$ AND $r$ divide $x-y$ and are distinct primes, we have $pqr \mid x-y$