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Question Number 157839 by gsk2684 last updated on 28/Oct/21

find the last four digits of 11^(15999) ?

$${find}\:{the}\:{last}\:{four}\:{digits}\:{of}\: \\ $$$$\mathrm{11}^{\mathrm{15999}} ? \\ $$

Answered by Rasheed.Sindhi last updated on 29/Oct/21

11^(15999) ≡x(mod 10^4 ) ∵ gcd(11,10^4 )=1 ∴ 11^(φ(10^4 )) ≡1(mod10^4 ) φ(10^4 )=4000 Least number n for which 11^n ≡1(mod 10000) must be any divisor of φ(10^4 )=4000 Trying for n=1,2,4,...,500,800,...4000 we can see that 11^(500) ≡1(mod10^4 ) ∵15999=500×32−1 (11^(500) )^(32) ≡(1)^(32) (mod10^4 ) 11^(16000) ≡1(mod10^4 ) 11^(16000) ≡1+10^4 ×10(mod10^4 ) 11^(16000) ≡100001(mod10^4 ) Dividing by 11 11^(15999) ≡9091(mod10^4 ) Last 4 digits of 11^(5999) are: 9091

$$\mathrm{11}^{\mathrm{15999}} \equiv{x}\left({mod}\:\mathrm{10}^{\mathrm{4}} \right) \\ $$$$\because\:{gcd}\left(\mathrm{11},\mathrm{10}^{\mathrm{4}} \right)=\mathrm{1} \\ $$$$\therefore\:\mathrm{11}^{\phi\left(\mathrm{10}^{\mathrm{4}} \right)} \equiv\mathrm{1}\left({mod}\mathrm{10}^{\mathrm{4}} \right) \\ $$$$\phi\left(\mathrm{10}^{\mathrm{4}} \right)=\mathrm{4000} \\ $$$${Least}\:{number}\:{n}\:{for}\:{which} \\ $$$$\:\:\:\:\mathrm{11}^{{n}} \equiv\mathrm{1}\left({mod}\:\mathrm{10000}\right) \\ $$$${must}\:{be}\:{any}\:{divisor}\:{of}\:\phi\left(\mathrm{10}^{\mathrm{4}} \right)=\mathrm{4000} \\ $$$$ \\ $$$$\mathcal{T}{rying}\:{for}\:{n}=\mathrm{1},\mathrm{2},\mathrm{4},...,\mathrm{500},\mathrm{800},...\mathrm{4000} \\ $$$${we}\:{can}\:{see}\:{that} \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{11}^{\mathrm{500}} \equiv\mathrm{1}\left({mod}\mathrm{10}^{\mathrm{4}} \right) \\ $$$$\because\mathrm{15999}=\mathrm{500}×\mathrm{32}−\mathrm{1} \\ $$$$\:\:\:\:\:\:\:\:\:\:\left(\mathrm{11}^{\mathrm{500}} \right)^{\mathrm{32}} \equiv\left(\mathrm{1}\right)^{\mathrm{32}} \left({mod}\mathrm{10}^{\mathrm{4}} \right) \\ $$$$\:\:\:\:\:\:\:\mathrm{11}^{\mathrm{16000}} \equiv\mathrm{1}\left({mod}\mathrm{10}^{\mathrm{4}} \right) \\ $$$$\:\:\:\:\:\:\mathrm{11}^{\mathrm{16000}} \equiv\mathrm{1}+\mathrm{10}^{\mathrm{4}} ×\mathrm{10}\left({mod}\mathrm{10}^{\mathrm{4}} \right) \\ $$$$\:\:\:\:\:\:\mathrm{11}^{\mathrm{16000}} \equiv\mathrm{100001}\left({mod}\mathrm{10}^{\mathrm{4}} \right) \\ $$$${Dividing}\:{by}\:\mathrm{11} \\ $$$$\:\:\:\:\:\:\mathrm{11}^{\mathrm{15999}} \equiv\mathrm{9091}\left({mod}\mathrm{10}^{\mathrm{4}} \right) \\ $$$$\:{Last}\:\mathrm{4}\:{digits}\:{of}\:\mathrm{11}^{\mathrm{5999}} \:{are}:\:\mathrm{9091} \\ $$

Commented by gsk2684 last updated on 31/Oct/21

thanks

$${thanks}\: \\ $$

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