Cyclicity:To find the units digit of a number

Question

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

Answer 1

$33^4\equiv 3^4\equiv 81\equiv 1\pmod{10}$.

$34^{35^{36^{37^{38}}}}=4k$ 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.

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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 $1$.