modular arithmetic - Calculate a number (mod)

Calculate: $3^{1234} $ (mod 17)

We're not suppose to use any "tricks" like the little theorem or anything else alike because we haven't learned that yet just the the definition of modulo.

I tried to do this but it doesn't really help:

$3^{1234}=20^{1234}=2^{1234}10^{1234} $

Thanks in advance.

Answer

Doing arithmetic modulo $\;17\;$ all along:

$$3^4=81=-4\;,\;\;3^5=-12=5\;,\;\;3^6=15=-2\;,\;\;3^7=-6\;,\;\;3^8=-18=-1\implies$$

$$\implies 3^{16}=1\;,\;\;\text{and $\;3\;$ is a primitive root modulo}\;17$$

Now:

$$1234=77\cdot 16+2\implies3^{1234}=(3^{16})^{77}\cdot3^2=3^2=9$$