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Question Number 215951 by alcohol last updated on 22/Jan/25

$$E_n=3^{E_{n-1}},n⩾2$$$$findtheunitdigitof{E}_{1000}$$

Answered by A5T last updated on 22/Jan/25

$$3^{3^{3^{\mathrm{x}}}}\equiv 7\left(\mathrm{mod}10\right)$$$$3^{3^{\mathrm{x}}}\equiv {(-1)}^{3^{\mathrm{x}}}\equiv -1\left(\mathrm{mod}4\right)$$$$\Rightarrow 3^{3^{\mathrm{x}}}=4\mathrm{k}+3$$$$\Rightarrow 3^{3^{3^{\mathrm{x}}}}\equiv 3^{4\mathrm{k}}\times 3^3={81}^{\mathrm{k}}\times 27\equiv 1\times 7=7\left(\mathrm{mod}10\right)$$$$\Rightarrow {\mathrm{E}}_{1000}\equiv 7\left(\mathrm{mod}10\right)$$

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