Question

chứng minh rằng \(A=5^{n+2}+5^{n+1}+5^n\) chia hết cho 31

Answer 1

\(A=5^{n+2}+5^{n+1}+5^n\)

\(=5^n.5^2+5^n.5^1+5^n.1\)  (tách lũy thừa thành tích) 

\(=5^n\left(5^2+5^1+1\right)=5^n.31⋮31^{\left(dpcm\: \right)}\) (tách ra thừa số chung)

Answer 2

\(A=5^{n+2}+5^{n+1}+5^n=5^n.\left(5^2+5^1+1\right)=5^n.\left(25+5+1\right)=31.5^n⋮31\)

Answer 3

\(5^{n+2}\)+\(5^{n+1}\)+\(5^n\)= \(5^n\)x\(5^2\)+\(5^n\)x5+\(5^n\)
                                    = \(5^n\)x(\(5^2\)+5+1)
                                  = \(5^n\)x31( chia hết cho 31) ( đpcm)