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What is the remainder when 333^{333} is divided by 13?

 Apr 19, 2022
 #1
avatar+115 
-3

I believe that your answer would be 25 and a remainder of 8

sincerely ▄︻デ✞☬🇮‌🇫‌1️⃣🇳‌1️⃣🇹‌¥☬✞══━一

 Apr 19, 2022
 #2
avatar+9459 
+1

Note that \(\varphi(13) = 13 - 1 = 12\), where \(\varphi\) denotes the Euler totient function.

 

Now, for these type of problems, you can take mod 13 of the base and mod \(\varphi(13)\) of the power.

 

\(\begin{array}{rcl} 333^{333} &\equiv& 8^9 \pmod{13}\\ &=& 2^{27} \pmod{13}\\ &=& 2^3 \pmod{13} \end{array}\)

 

Note that I used \(8^9 = (2^3)^9 = 2^{27}\) and then took mod \(\varphi(13)\) of the power again.

 

Now the size of the problem becomes manageable, because you can calculate 2^3, and that's the answer.

 Apr 19, 2022

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