Solve this problems with the Euler's theorem

Question

I've solved many of these examples, but I couldn't solve these problems:

I would be very thankful if you would solve this problems. Thank you very much for your help.

Answer 1

First problem is by Fermat: $123^{123}\equiv 2^{123}=(2^{10})^{12}2^3\equiv2^3\equiv 8 \bmod 11$

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Second problem by Euler, using $\varphi(27)=18$:

$10^{188}= ((10)^{18})^{10}10^8\equiv 10^8 \bmod 27$

we now evaluate this as $((10^2)^2)^2\equiv((19)^2)^2\equiv(10)^2\equiv19 \bmod 27$

Note the second problem can also be solved by noting $10^3\equiv 1 \bmod 27$ but this cannot be obtained by Euler.

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Third problem is by Euler using $\varphi(44)=20$:

$(4447)^{2018}\equiv 3^{2018}=(3^{20})^{100}3^{18}\equiv 3^{18} \bmod 44$

We now evaluate this as $(3^6)^3\equiv(729)^3\equiv(25)^3\equiv 5 \bmod 44$