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Question Number 106596 by mathdave last updated on 06/Aug/20

Answered by john santu last updated on 06/Aug/20

$$@\mathrm{JS}@$$$$(2^{\mathrm{x}}-1)(\mathrm{x}+3)(\mathrm{x}-1)=0$$$$\to \{\begin{array}{l}\mathrm{x}=-3;\mathrm{x}=1;\mathrm{x}=0\}\end{array}$$$$$$

Answered by Rasheed.Sindhi last updated on 06/Aug/20

$$3.4^n+51$$$${}^{\bullet }3\mid 3.4^n\mathrm{\&}3\mid 51\Rightarrow 3\mid (3.4^n+51)$$$$.....$$

Answered by JDamian last updated on 06/Aug/20

$$\left(1\right)$$$$\frac{3\cdot 4^n+51}{3}=4^n+17$$$$(4^n+17)mod3=\left(4^{n}mod3\right)+\left(17mod3\right)=$$$$={\left(4mod3\right)}^n+2=1^n+2=3mod3=0$$

Answered by 1549442205PVT last updated on 06/Aug/20

$$\mathrm{We}\mathrm{have}:$$$$4^{\mathrm{n}}={(3+1)}^{\mathrm{n}}=3^{\mathrm{n}}+{\mathrm{C}}_{\mathrm{n}}^1.3^{\mathrm{n}-1}+{\mathrm{C}}_{\mathrm{n}}^2.3^{\mathrm{n}-2}+...$$$$+...{\mathrm{C}}^{\mathrm{n}-1}.3+1=3\mathrm{m}+1.$$$$\Rightarrow 4^{\mathrm{n}}+17=3\mathrm{m}+1+17=3\mathrm{m}+18=3(\mathrm{m}+6)$$$$\Rightarrow 3.4^{\mathrm{n}}+51=3(4^{\mathrm{n}}+17)=9(\mathrm{m}+6)⋮9$$$$\mathrm{Consequently},3.4^{\mathrm{n}}+51\mathrm{is}\mathrm{divisible}\mathrm{by}$$$$\mathrm{both}3\mathrm{and}9(\mathrm{q}.\mathrm{e}.\mathrm{d})$$

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