Is $a^x \pmod{n} = (a\pmod{n})^{x \pmod{n}}$? [closed]
Solution 1:
Not necessarily. Note that
$$a^{x \text{ mod }n} \text{mod }n \equiv a^x \text{mod } n$$ implies that
$$a^n \text{ mod n}\equiv 1$$,
which by Fermat's Little Theorem is wrong in infinitely many cases, as if n is prime,
$$a^n \text{ mod n} \equiv a^{n-1}$$ which is almost never true. An example of this is $a=2,n=7,x=8$, resulting in
$$2 \text{ mod 7} \equiv 2^8 \text{ mod 7}\equiv 4$$
Which clearly isn't true