<s>[INST] Cyclicity:To find the units digit of a number 
Can you please help me to find the units digit of this number:
  33^34^35^36^37^38 .. I got this question in CSR 2017

 [/INST] $33^4\equiv 3^4\equiv 81\equiv 1\pmod{10}$.
$34^{35^{36^{37^{38}}}}=4k$ for some $k\in\mathbb Z^+$.
$33^{4k}\equiv \left(33^4\right)^k\equiv 1^k\equiv 1\pmod{10}$.
Euler's theorem is relevant here, but I've been able to explain this simply without it.

Edit: The multiplicative order of $33$ modulo $10$ is $4$.
You could use Euler's theorem to notice that $\phi(10)=4$ and continue.
To explain it more simply, notice the pattern:
$33^0\equiv 1\pmod{10}$
$33^1\equiv 3\pmod{10}$
$33^2\equiv 3^2\equiv 9\pmod{10}$
$33^3\equiv 3^3\equiv 27\equiv 7\pmod{10}$
$33^4\equiv 3^4\equiv 81\equiv 1\pmod{10}$
$33^5\equiv 33\cdot 33^4\equiv 33\cdot 1\equiv 3\pmod{10}$
...
Since $34^{35^{36^{37^{38}}}}$ is divisible by $4$, we get the answer $1$.
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