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Question Number 41720 by Tawa1 last updated on 11/Aug/18
Difference of last two digits of 2019^(2019^(2019) ) is ?
$$\mathrm{Difference}\:\mathrm{of}\:\mathrm{last}\:\mathrm{two}\:\mathrm{digits}\:\mathrm{of}\:\:\:\:\mathrm{2019}^{\mathrm{2019}^{\mathrm{2019}} } \:\:\:\mathrm{is}\:\:? \\ $$
Commented by MJS last updated on 12/Aug/18
=2019×2019×2019×... so we first need to know the sequence of the last 2 digits of 19^n : {19, 61, 59, 21, 99, 81, 39, 41, 79, 01, ...} the difference between these last 2 digits: {8, 5, 4, 1, 0, 7, 6, 3, 2, 1, ...} both have a period of 10 ⇒ we need to know the remainder of ((2019^(2019) )/(10)) the sequence of the remainders of ((2019^n )/(10)) {9, 1, 9, 1, ...} so the remainder is 9 if n is uneven and 1 if n is even 2019 is uneven ⇒ remainder is 9 ⇒ last two digits of 2019^(2019^(2019) ) =79 ⇒ difference is 2
$$=\mathrm{2019}×\mathrm{2019}×\mathrm{2019}×… \\ $$$$\mathrm{so}\:\mathrm{we}\:\mathrm{first}\:\mathrm{need}\:\mathrm{to}\:\mathrm{know}\:\mathrm{the}\:\mathrm{sequence}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{last}\:\mathrm{2}\:\mathrm{digits}\:\mathrm{of}\:\mathrm{19}^{{n}} : \\ $$$$\left\{\mathrm{19},\:\mathrm{61},\:\mathrm{59},\:\mathrm{21},\:\mathrm{99},\:\mathrm{81},\:\mathrm{39},\:\mathrm{41},\:\mathrm{79},\:\mathrm{01},\:…\right\} \\ $$$$\mathrm{the}\:\mathrm{difference}\:\mathrm{between}\:\mathrm{these}\:\mathrm{last}\:\mathrm{2}\:\mathrm{digits}: \\ $$$$\left\{\mathrm{8},\:\mathrm{5},\:\mathrm{4},\:\mathrm{1},\:\mathrm{0},\:\mathrm{7},\:\mathrm{6},\:\mathrm{3},\:\mathrm{2},\:\mathrm{1},\:…\right\} \\ $$$$ \\ $$$$\mathrm{both}\:\mathrm{have}\:\mathrm{a}\:\mathrm{period}\:\mathrm{of}\:\mathrm{10}\:\Rightarrow \\ $$$$\mathrm{we}\:\mathrm{need}\:\mathrm{to}\:\mathrm{know}\:\mathrm{the}\:\mathrm{remainder}\:\mathrm{of}\:\frac{\mathrm{2019}^{\mathrm{2019}} }{\mathrm{10}} \\ $$$$\mathrm{the}\:\mathrm{sequence}\:\mathrm{of}\:\mathrm{the}\:\mathrm{remainders}\:\mathrm{of}\:\frac{\mathrm{2019}^{{n}} }{\mathrm{10}} \\ $$$$\left\{\mathrm{9},\:\mathrm{1},\:\mathrm{9},\:\mathrm{1},\:…\right\} \\ $$$$\mathrm{so}\:\mathrm{the}\:\mathrm{remainder}\:\mathrm{is}\:\mathrm{9}\:\mathrm{if}\:{n}\:\mathrm{is}\:\mathrm{uneven}\:\mathrm{and} \\ $$$$\mathrm{1}\:\mathrm{if}\:{n}\:\mathrm{is}\:\mathrm{even} \\ $$$$\mathrm{2019}\:\mathrm{is}\:\mathrm{uneven}\:\Rightarrow\:\mathrm{remainder}\:\mathrm{is}\:\mathrm{9}\:\Rightarrow\:\mathrm{last}\:\mathrm{two} \\ $$$$\mathrm{digits}\:\mathrm{of}\:\mathrm{2019}^{\mathrm{2019}^{\mathrm{2019}} } \:=\mathrm{79}\:\Rightarrow\:\mathrm{difference}\:\mathrm{is}\:\mathrm{2} \\ $$
Commented by Tawa1 last updated on 15/Aug/18
God bless you sir
$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir} \\ $$

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