Modular arithmetic problem (mod $13$)

modular-arithmetic

$$\large2017^{(2020^{2015})} \pmod{13}$$

I am practicing for my exam and I can solve almost all problem, but this type of problem is very hard to me. In this case, I have to compute this by modulo $13$.


Solution 1:

As $2017\equiv2\pmod{13},$

$$2017^{2020^{2015}}\equiv2^{2020^{2015}}\pmod{13}$$

Now as $(2020,\phi(13))=4,$ let us find $2020^{2015-1}\pmod{\dfrac{12}4}$

As $2020\equiv1\pmod3\implies2020^{2015-1}\equiv1\pmod3$

$2020^{2015}\equiv2020\cdot1\pmod{3\cdot2020}\equiv2020\pmod{12}\equiv4$

$\implies2^{2020^{2015}}\equiv2^4\pmod{13}=?$

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