RSA Encryption for specitic messages x with x = ap mod pq for ap-bq=1
You can show that $ap\mod q =(ap)^e\mod q$ by using the fact, that $ap-bq=1\Longleftrightarrow ap=bq+1$.
You know that $q|bq$ and therefor $ap\mod q = bq+1\mod q \equiv 1\mod q$.
This implicates that $(ap)^e\mod q = (bq+1)^e\mod q \equiv 1\mod q$. Therefor $ap\mod q \equiv (ap)^e\mod q.$
Now with the chinsese remainder theorem we know that $ap\equiv_N(ap)^e$.