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Question Number 84739 by Rio Michael last updated on 15/Mar/20

$$\mathrm{find}\mathrm{the}\mathrm{unit}\mathrm{digit}\mathrm{in}\mathrm{the}\mathrm{number}$$$${15}^{1789}+{17}^{1789}+{19}^{1789}$$

Answered by TANMAY PANACEA last updated on 15/Mar/20

$$lastdigitof{15}^{1789}=5$$$${17}^1\to lastdigit=7$$$${17}^2\to lastdigit=9$$$${17}^3\to lastdigit=3$$$${17}^4\to lastdigit=1$$$$sorepetationof7931cyclicway$$$$1789/4=447+\frac{1}{4}$$$$\mathit{s}\mathit{o}\mathit{l}\mathit{a}\mathit{s}\mathit{t}\mathit{d}\mathit{i}\mathit{g}\mathit{i}\mathit{t}\mathit{o}\mathit{f}{17}^{1789}=7$$$${19}^{1789}$$$${19}^1\to l.d=9$$$${19}^2\to l.d=1$$$$cyclicof9,1$$$$lastdigitof{19}^{1789}$$$$1789/2=894+\frac{1}{2}$$$$lastdigitof{19}^{1789}is=9$$$$so$$$$lastdigitof{15}^{1789}+{17}^{1789}+{19}^{1789}$$$$is(5+7+9=21$$$$so\mathit{l}\mathit{a}\mathit{s}\mathit{t}\mathit{d}\mathit{i}\mathit{g}\mathit{i}\mathit{t}\mathit{o}\mathit{f}\mathit{g}\mathit{i}\mathit{v}\mathit{e}\mathit{n}\mathit{e}\mathit{x}\mathit{o}\mathit{r}\mathit{e}\mathit{s}\mathit{s}\mathit{i}\mathit{o}\mathit{n}=1$$

Commented by Rio Michael last updated on 15/Mar/20

$$\mathrm{thanks}\mathrm{so}\mathrm{much}\mathrm{very}\mathrm{clear}$$

Commented by mr W last updated on 15/Mar/20

$$whatarethelast\mathit{t}\mathit{w}\mathit{o}digits?$$

Commented by Rio Michael last updated on 15/Mar/20

$$\mathrm{from}\mathrm{the}\mathrm{calculation}\mathrm{and}\mathrm{my}\mathrm{understanding}$$$$\mathrm{it}\mathrm{will}\mathrm{be}\mathrm{the}\mathrm{sum}\mathrm{of}\mathrm{the}\mathrm{last}\mathrm{two}\mathrm{digits}\mathrm{of}$$$$\mathrm{all}\mathrm{three}.$$

Commented by mr W last updated on 15/Mar/20

$${that}^{\prime }sclear.butwhatarethey?$$

Commented by Rio Michael last updated on 15/Mar/20

$$\mathrm{the}\mathrm{last}\mathrm{two}\mathrm{digits}\mathrm{of}\mathrm{the}\mathrm{number}1235432674\mathrm{is}7\mathrm{and}4\mathrm{respectively}$$$$\mathrm{this}\mathrm{can}\mathrm{be}\mathrm{gotten}\mathrm{when}\mathrm{this}\mathrm{number}\mathrm{is}\mathrm{divided}\mathrm{by}100\mathrm{and}\mathrm{the}$$$$\mathrm{remainder}\mathrm{noted}.$$$$$$

Commented by mr W last updated on 15/Mar/20

$${that}^{\prime }salsoclear.butcanyougetthe$$$$lasttwodigitsfrom{15}^{1789}and{17}^{1789}$$$$and{19}^{1789}respectively?$$

Commented by Rio Michael last updated on 15/Mar/20

$$\mathrm{that}\mathrm{will}\mathrm{be}\mathrm{difficult}\mathrm{to}\mathrm{do}$$

Commented by mr W last updated on 15/Mar/20

$$igot$$$$lasttwodigitsfrom{15}^{1789}are75$$$$lasttwodigitsfrom{17}^{1789}are97$$$$lasttwodigitsfrom{19}^{1789}are79$$$$\Rightarrow lasttwodigitsfrom{15}^{1789}+{17}^{1789}+{19}^{1789}$$$$are51.$$

Commented by Rio Michael last updated on 15/Mar/20

$$\mathrm{wow}\mathrm{sir}\mathrm{thats}\mathrm{great}\mathrm{work}$$

Commented by mr W last updated on 16/Mar/20

$$seeQ84782$$

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