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Question Number 158906 by SANOGO last updated on 10/Nov/21

determiner le reste de la division eucludienne de:  10^(100)  par 105

$${determiner}\:{le}\:{reste}\:{de}\:{la}\:{division}\:{eucludienne}\:{de}: \\ $$$$\mathrm{10}^{\mathrm{100}} \:{par}\:\mathrm{105} \\ $$

Answered by Rasheed.Sindhi last updated on 11/Nov/21

                     Mod 105  10^(100) =(10^2 )^(50) ≡(−5)^(50) =(5^4 )^(12) .5^2   ≡(−5)^(12) .5^2 =(5^2 )^6 .5^2 =5^(14) =(5^4 )^3 .5^2   ≡(−5)^3 .5^2 =(−5)^5 =(5^4 )(−5)  ≡(−5)(−5)=25            Used congruences         determinant (((     10^2 =100≡−5(mod 105)    )))    determinant (((5^4 =25^2 =625≡−5(mod 105))))

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\underline{\mathrm{Mod}\:\mathrm{105}} \\ $$$$\mathrm{10}^{\mathrm{100}} =\left(\mathrm{10}^{\mathrm{2}} \right)^{\mathrm{50}} \equiv\left(−\mathrm{5}\right)^{\mathrm{50}} =\left(\mathrm{5}^{\mathrm{4}} \right)^{\mathrm{12}} .\mathrm{5}^{\mathrm{2}} \\ $$$$\equiv\left(−\mathrm{5}\right)^{\mathrm{12}} .\mathrm{5}^{\mathrm{2}} =\left(\mathrm{5}^{\mathrm{2}} \right)^{\mathrm{6}} .\mathrm{5}^{\mathrm{2}} =\mathrm{5}^{\mathrm{14}} =\left(\mathrm{5}^{\mathrm{4}} \right)^{\mathrm{3}} .\mathrm{5}^{\mathrm{2}} \\ $$$$\equiv\left(−\mathrm{5}\right)^{\mathrm{3}} .\mathrm{5}^{\mathrm{2}} =\left(−\mathrm{5}\right)^{\mathrm{5}} =\left(\mathrm{5}^{\mathrm{4}} \right)\left(−\mathrm{5}\right) \\ $$$$\equiv\left(−\mathrm{5}\right)\left(−\mathrm{5}\right)=\mathrm{25} \\ $$$$\:\:\:\:\:\:\:\:\:\:\underline{\mathrm{Used}\:\mathrm{congruences}}\:\:\:\:\:\: \\ $$$$\begin{array}{|c|}{\:\:\:\:\:\mathrm{10}^{\mathrm{2}} =\mathrm{100}\equiv−\mathrm{5}\left({mod}\:\mathrm{105}\right)\:\:\:\:}\\\hline\end{array}\: \\ $$$$\begin{array}{|c|}{\mathrm{5}^{\mathrm{4}} =\mathrm{25}^{\mathrm{2}} =\mathrm{625}\equiv−\mathrm{5}\left({mod}\:\mathrm{105}\right)}\\\hline\end{array}\: \\ $$

Commented by Rasheed.Sindhi last updated on 12/Nov/21

You′re welcome sir!

$${You}'{re}\:{welcome}\:{sir}! \\ $$

Commented by Rasheed.Sindhi last updated on 12/Nov/21

Why did you delete your answer sir?  It would really require some   minor changes.The logic was  very correct!

$${Why}\:{did}\:{you}\:{delete}\:{your}\:{answer}\:{sir}? \\ $$$${It}\:{would}\:{really}\:{require}\:{some} \\ $$$$\:\boldsymbol{{minor}}\:{changes}.\mathcal{T}{he}\:{logic}\:{was} \\ $$$${very}\:{correct}! \\ $$

Commented by SANOGO last updated on 11/Nov/21

merci bien

$${merci}\:{bien} \\ $$

Commented by puissant last updated on 12/Nov/21

Yes sir my logic was correct but i did  not achieve the result is proved that  your logic is clearer really thank you   very much sir..

$${Yes}\:{sir}\:{my}\:{logic}\:{was}\:{correct}\:{but}\:{i}\:{did} \\ $$$${not}\:{achieve}\:{the}\:{result}\:{is}\:{proved}\:{that} \\ $$$${your}\:{logic}\:{is}\:{clearer}\:{really}\:{thank}\:{you}\: \\ $$$${very}\:{much}\:{sir}.. \\ $$

Commented by puissant last updated on 12/Nov/21

Thanks sir...

$${Thanks}\:{sir}... \\ $$

Commented by Rasheed.Sindhi last updated on 12/Nov/21

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