A property of twin primes

Question

$(a,b)$ is a couple of twin primes such that $b=a+2$ and $a > 29$.
Let $N = 4^b$ and $q$ the quotient which results from the division of $N$ by $a$ and $r$ is the remainder.
We calculate $P = (q\bmod b)a+r-1$
Below we prove that $P = 3(10b+1)$ using Fermat's little theorem.

$N=16\cdot4^a\equiv64\pmod a$ then $r=64$ ($a > 64$)
$4^b=(b-2)q+64$ then $4\equiv64-2q\pmod b$ and $q\equiv30\pmod b$
Finally we have $P=30a+63=3(10b+1)$

My question is :

1. Is the proof correct?
2. I was wondering if there is another way to proof this property?

Thanks.