Question

Chứng minh rằng 70.27^1001 + 31.38^101 chia hết cho 13.

Answer 1

Ta có 

\(70\equiv 5 (\mod 13);27\equiv 1(\mod 13)\Rightarrow 27^{101}\equiv 1(\mod 13)\\\Rightarrow 70.27^{101}\equiv 5(\mod 13)\\ 31\equiv 5 (\mod 13);38^2\equiv 1(\mod 13)\Rightarrow (38^{2})^{50}.38\equiv 38\equiv 12 (\mod 13)\\\Rightarrow 31.38^{101}\equiv 8 (\mod 13) \Rightarrow 70.27^{101}+31.38^{101}\equiv 13(\mod 13)\)

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