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Question Number 55787 by gunawan last updated on 04/Mar/19

$$\mathrm{The}\mathrm{remainder}\mathrm{when}{5}^{99}\mathrm{is}\mathrm{divided}\mathrm{by}13\mathrm{is}$$

Answered by MJS last updated on 04/Mar/19

$$\mathrm{remainders}\mathrm{of}\frac{5^n}{13}$$$$n=1+4k:5$$$$n=2+4k:12$$$$n=3+4k:8$$$$n=4+4k:3$$$$99=3+4\times 24\Rightarrow \mathrm{remainder}\mathrm{is}8$$

Answered by tanmay.chaudhury50@gmail.com last updated on 04/Mar/19

$$tryingotherway$$$$5^{99}$$$$={\left(5^3\right)}^{33}$$$$={\left(125\right)}^{33}$$$$={(9\times 13+8)}^{33}$$$$={\left(9\times 13\right)}^{33}+33c_1{\left(9\times 13\right)}^{32}\times 8+...+8^{33}$$$$=f\left(13\right)+{\left(512\right)}^{11}$$$$=f\left(13\right)+{(13\times 39+5)}^{11}$$$$=f\left(13\right)+g\left(13\right)+5^{11}$$$$=f(13+g\left(13\right)+5\times {\left(5^2\right)}^5$$$$=f\left(13\right)+g\left(13\right)+5\times {(13+12)}^5$$$$=f\left(13\right)+g\left(13\right)+5\times [{13}^5+5c_1{13}^4+...+{12}^5]$$$$=f\left(13\right)+g\left(13\right)+5\times h\left(13\right)+5\times {12}^5$$$$=f\left(13\right)+g\left(13\right)+5\times h\left(13\right)+5\times {\left(144\right)}^3$$$$=f\left(13\right)+g\left(13\right)+5\times h\left(13\right)+5\times {(13\times 11+1)}^3$$$$=f\left(13\right)+g\left(13\right)+5\times h\left(13\right)+5\times [{\left(13\times 11\right)}^3+3\times {\left(13\times 11\right)}^2+3\times \left(13\times 11\right)\times 1^2+1^3]$$$$=f\left(13\right)+g\left(13\right)+5\times h\left(13\right)+5\times \gamma \left(13\right)+5\times 1^3$$$$soinmycalculationremaideris5$$

Commented by mr W last updated on 05/Mar/19

$$hereiswrongsir!$$$$=f\left(13\right)+g\left(13\right)+5\times h\left(13\right)+5\times {12}^5$$$$=f\left(13\right)+g\left(13\right)+5\times h\left(13\right)+5\times {\left(144\right)}^3$$$$itshouldbe:$$$$=f\left(13\right)+g\left(13\right)+5\times h\left(13\right)+5\times 12\times {\left(144\right)}^2$$$$=f\left(13\right)+g\left(13\right)+5\times h\left(13\right)+60\times {(13\times 11+1)}^2$$$$=......+60$$$$=......+13\times 4+8$$$$=......+8$$$$\Rightarrow remainderis8.$$

Commented by tanmay.chaudhury50@gmail.com last updated on 05/Mar/19

$$thankyousir..manymanythanks...$$

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