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Question Number 195443 by MM42 last updated on 02/Aug/23

$${10}^{10}+{10}^{{10}^2}+{10}^{{10}^3}+...+{10}^{{10}^{10}}\stackrel{7}{\equiv }?$$

Answered by BaliramKumar last updated on 02/Aug/23

$$5$$

Commented by MM42 last updated on 02/Aug/23

$$ok$$

Answered by BaliramKumar last updated on 02/Aug/23

$$\varphi \left(7\right)=6$$$$\frac{{10}^{\mathrm{N}}}{6}\equiv 4$$$${10}^{10}+{10}^{{10}^2}+{10}^{{10}^3}+...+{10}^{{10}^{10}}\stackrel{7}{\equiv }?$$$${10}^4+{10}^4+...............\left(10\mathrm{time}\right)\stackrel{7}{\equiv }?$$$$3^4+3^4+...............\left(10\mathrm{time}\right)\stackrel{7}{\equiv }?$$$$3^4\times 10\stackrel{7}{\equiv }?$$$$3^4\times 3\stackrel{7}{\equiv }?$$$$3^5\stackrel{7}{\equiv }?$$$$9\times 9\times 3\stackrel{7}{\equiv }?$$$$2\times 2\times 3\stackrel{7}{\equiv }?$$$$12\stackrel{7}{\equiv }?$$$$5$$

Commented by MM42 last updated on 03/Aug/23

$$yoursolutionwasbeautiful$$$$anothersolution$$$${10}^n=3k+1;‘‘k^{″}isodd.$$$$10\underset{7}{\equiv }3$$$${10}^{10}+{10}^{{10}^2}+{10}^{{10}^3}+...+{10}^{{10}^{10}}$$$$\underset{7}{\equiv }{3}^{10}+3^{{10}^2}+3^{{10}^3}+...+3^{{10}^{10}}$$$$=3^{3k_1+1}+3^{3k_2+1}+3^{3k_3+1}+...+3^{3k_{10}+1};‘‘k_i^{″}isodd$$$$=3\times {\left(27\right)}^{k_1}+3\times {\left(27\right)}^{k_2}+3\times {\left(27\right)}^{k_3}+...+3\times {\left(27\right)}^{k_{10}}$$$$\underset{7}{\equiv }-3-3-3-...-3=-30\underset{7}{\equiv }5$$$$$$

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