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Question Number 7859 by tawakalitu last updated on 21/Sep/16

Find the remainder if   49^(1296)  × 7^(131)   is divided  by  13

$${Find}\:{the}\:{remainder}\:{if}\:\:\:\mathrm{49}^{\mathrm{1296}} \:×\:\mathrm{7}^{\mathrm{131}} \:\:{is}\:{divided} \\ $$$${by}\:\:\mathrm{13}\:\: \\ $$

Answered by sandy_suhendra last updated on 21/Sep/16

49^(1296)  × 7^(131)  = (7^2 )^(1296)  × 7^(131)   = 7^(2592)  × 7^(131)  = 7^(2723)   7^n  ÷ 13 have the same remainder  with 7^(n+12) ÷13  2723÷12=226 remainder 11  remainder of  7^(2723)  ÷ 13   is the same remainder 7^(11) ÷13  so the reminder is 2

$$\mathrm{49}^{\mathrm{1296}} \:×\:\mathrm{7}^{\mathrm{131}} \:=\:\left(\mathrm{7}^{\mathrm{2}} \right)^{\mathrm{1296}} \:×\:\mathrm{7}^{\mathrm{131}} \\ $$$$=\:\mathrm{7}^{\mathrm{2592}} \:×\:\mathrm{7}^{\mathrm{131}} \:=\:\mathrm{7}^{\mathrm{2723}} \\ $$$$\mathrm{7}^{{n}} \:\boldsymbol{\div}\:\mathrm{13}\:{have}\:{the}\:{same}\:{remainder} \\ $$$${with}\:\mathrm{7}^{{n}+\mathrm{12}} \boldsymbol{\div}\mathrm{13} \\ $$$$\mathrm{2723}\boldsymbol{\div}\mathrm{12}=\mathrm{226}\:{remainder}\:\mathrm{11} \\ $$$${remainder}\:{of}\:\:\mathrm{7}^{\mathrm{2723}} \:\boldsymbol{\div}\:\mathrm{13}\: \\ $$$${is}\:{the}\:{same}\:{remainder}\:\mathrm{7}^{\mathrm{11}} \boldsymbol{\div}\mathrm{13} \\ $$$${so}\:{the}\:{reminder}\:{is}\:\mathrm{2} \\ $$

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