What is the remainder of 333^444 + 444^333 divided by 7?

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2 Answers
Narad T.
Jan 5, 2018

The remainder is #=0#

Explanation:

Perform this by the arithmetic congruency modulo #7#

#"first part"#

#111≡6[7]#

#333≡18≡4[7]#

#4^2≡2[7]#

#4^3≡1[7]#

Therefore,

#333^444≡4^444[7]≡(4^3)^148≡1^148≡1[7]#

#"second part"#

#111≡6[7]#

#444≡24≡3[7]#

#3^2≡2[7]#

#3^3≡-1[7]#

Therefore,

#444^333≡(3)^333[7]≡((3)^111)^3≡(-1)^3≡-1[7]#

Finally,

#333^444+444^333≡1-1≡0[7]#

Cem Sentin
Jan 5, 2018

#333^444+444^333=0 (Mod 7)#

Explanation:

#333=4 (Mod 7)#

#333^2=4^2=2 (Mod 7)#

#333^3=4^3=1 (Mod 7)#

Due to #444=0 (Mod 3)#, #333^444=3^0=1 (Mod 7)#

#444=3 (Mod 7)#

#444^2=3^2=2 (Mod 7)#

#444^3=3^3=6 (Mod 7)#

#444^4=3^4=4 (Mod 7)#

#444^5=3^5=5 (Mod 7)#

#444^4=3^6=1 (Mod 7)#

Due to #333=3 (Mod 6)#, #444^333=3^3=6 (Mod 7)#

Thus,

#333^444+444^333=1+6=7=0 (Mod 7)#

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