What is the remainder when 50^51^52 divided by 11?

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To find the remainder:

$$50^{51^{52}} \mod 11$$

I have solved till:

$$6^{51^{52}} \mod 11$$

But not able to proceed further. Help please.

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2 Answers

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Use Euler's theorem. $50^{51^{52}}$ is going to be congruent to $50^a$ if $a$ and $51^{52}$ are congruent $\bmod \varphi (11)=10$. So we only need to calculate the last digit of $51^{52}$, which is $1$. Therefore $50^{51^{52}}\equiv 50^1\equiv 6 \bmod 11$

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$a^{b}\bmod{c}=a^{(b\bmod{\phi(c)})}\bmod{c}$

Therefore:

$50^{51^{52}}\bmod{11}=50^{(\color\red{51^{52}\bmod{\phi(11)}})}\bmod{11}=50^{(\color\red{51^{52}\bmod{10}})}\bmod{11}=50^{\color\red{1}}\bmod{11}=6$

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