How to solve $100^{63}$ mod 63

Question

I am trying to solve this question but not able to figure out how to approach it.

$100^{63} \mod\ {63}$

Please help.

Answer 1

Hint: $$100\equiv37 \mod 63$$ $$100^2\equiv46 \mod 63$$ $$100^3\equiv1 \mod 63$$ $$100^4\equiv37 \mod 63$$

Note: I don't know much about Modular arithmetic but I consulted Wolfram Alpha, and found this

Answer 2

Hint:

$100^3 \equiv 1 \pmod{63}$

Answer 3

As $\displaystyle63=9\cdot7$ and $(7,9)=1$

$\displaystyle100\equiv1\pmod9$

$\displaystyle100\equiv2\pmod7,\implies100^3\equiv2^3\equiv1$

$\displaystyle\implies100^{\text{lcm}(1,3)}\equiv1\pmod{7\cdot9}$

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Alternatively, using Carmichael function, $\displaystyle\lambda(63)=6\implies10^6\equiv1\pmod{63}$ as $(10,63)=1$

$\displaystyle\implies100^3=(10^2)^3=10^6\equiv1\pmod{63}$

In either case, $\displaystyle100^{3r}\equiv1\pmod{63}$ for any integer $r\ge0$