Question

Chung minh:

a/ \(5^5-5^4+5^3⋮\) 7

b/ \(7^6+7^5-7^4⋮11\)

c/ \(3^{x+3}+3^{x+1}+2^{x+3}+3^x-2^{x+1}⋮10\)

d/\(3^{x+3}+3^{x+1}+2^{x+3}+2^{x+2}⋮6\)

Answer 1

Ta có \(5^5-5^4+5^3=5^3\left(5^2-5+1\right)=5^3.21=5^3.3.7\)

Vì 5³.3 là số nguyên nên \(5^3.3.7⋮7\)

Vậy  \(5^5-5^4+5^3⋮7\)

Answer 2

c) \(3^{x+3}+3^{x+1}+2^{x+3}+2^{x+2}\)

\(=\left(3^{x+3}+3^{x+1}\right)+\left(2^{x+3}+2^{x+2}\right)\)

\(=3^x\left(3^2+3\right)+2^x\left(2^2+2\right)\)

\(=3^x.12+2^x.6\)

\(=6\left(2.3^x+2^x\right)\)

Vì \(2.3^x+2^x\in Z\)

Nên : \(6\left(2.3^x+2^x\right)⋮6\)

Vậy \(3^{x+3}+3^{x+1}+2^{x+3}+2^{x+2}⋮6\)