<s>[INST] How do you calculate $25^{11} \pmod{341}$? How do you calculate $25^{11} \pmod{341}$?
I understand you have to split the exponent into $11 = 1 + 2 + 8$?
 [/INST] Note that $ 341 = 11 \cdot 31 $ and hence it is sufficient to fine the residues of $ 25^{11} $ modulo $ 11 $ and $ 31 $ and then apply the Chinese Remainder Theorem.
By Fermat's Little Theorem, $ 25^{10} \equiv 1 \mod 11 \implies 25^{11} \equiv 25 \equiv 3 \mod 11 $. 
Also note that $ 25^4 \equiv (-6)^4 \equiv 5^2 \equiv 25 \mod 31 $ and hence $ 25^3 \equiv 1 \mod 31 $ which implies that $ 25^{11} \equiv 25^2 \equiv 5 \mod 31 $. 
Therefore, $ 25^{11} \equiv 36 \mod 341 $ after applying the CRT. 
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