what is the value of $2^{((p-n)*(q-n))}$ mod pq ? in which value of n varies...

Question

We know that:

$2^{(p-1)}$ mod $p =1$ [where p is prime]

and $2^{((p-1)*(q-1))}$ mod pq =1 [where p,q are coprimes]

what is the value of $2^{((p-n)*(q-n))}$ mod pq ? in which value of n varies...

Answer 1

The value of $2^{(p-n)(q-n))}\bmod pq$ can vary, so there is no "unique" value. Take, for example $(p,q)=(11,13)$ and $n=2$. Then $$ 2^{(p-2)(q-2)}\equiv 138\bmod pq. $$ For $n=3$ it is $$ 2^{(p-3)(q-3)}\equiv 100\bmod pq. $$ For $n=4$ it is $$ 2^{(p-4)(q-4)}\equiv 8\bmod pq. $$