Question

CMR:

a, 5^2005+5^2003 chia hết cho 13

b, a^2+b^2+1 lớn hơn hoặc bằng ab+a+b

c, cho a+b+c.CM:a^3+b^3+c^3=3abc

Answer 1

a) Ta có:

\(5^2=25\equiv-1\left(mod13\right)\)

\(\Rightarrow\left\{{}\begin{matrix}5^{2004}=\left(5^2\right)^{1002}\equiv\left(-1\right)^{1002}\left(mod13\right)\equiv1\left(mod13\right)\\5^{2002}=\left(5^2\right)^{1001}\equiv\left(-1\right)^{1001}\left(mod13\right)\equiv-1\left(mod13\right)\end{matrix}\right.\)

\(\Rightarrow\left\{{}\begin{matrix}5^{2005}=5^{2004}.5\equiv1.5\left(mod13\right)\equiv5\left(mod13\right)\\5^{2003}=5^{2002}.5\equiv\left(-1\right).5\left(mod13\right)\equiv-5\left(mod13\right)\end{matrix}\right.\)

\(\Rightarrow5^{2005}+5^{2003}\equiv5+\left(-5\right)\left(mod13\right)\equiv0\left(mod13\right)\)

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