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Question Number 63674 by Tawa1 last updated on 07/Jul/19

Show that the number 122^n − 102^n − 21^n is always one less than a multiple of 2020. For every positive integer n.

$$\mathrm{Show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{number}\:\:\mathrm{122}^{\mathrm{n}} \:−\:\mathrm{102}^{\mathrm{n}} \:−\:\mathrm{21}^{\mathrm{n}} \:\:\mathrm{is}\:\mathrm{always}\:\mathrm{one}\:\mathrm{less}\:\mathrm{than}\:\mathrm{a} \\ $$$$\:\mathrm{multiple}\:\mathrm{of}\:\:\mathrm{2020}.\:\:\mathrm{For}\:\mathrm{every}\:\mathrm{positive}\:\mathrm{integer}\:\:\mathrm{n}. \\ $$

Commented by Prithwish sen last updated on 07/Jul/19

(101+21)^n −(101+1)^n −21^n ={ 101^n +n101^(n−1) 21+......+n101.(21)^(n−1) +21^n }−{101^n +n101^(n−1) +.....+n101+1)−21^n = n101^(n−1) (21−1)+ ........+n101(21^(n−1) −1) −1 = 101×20{101^(n−2) +.......n(21^(n−2) +21^(n−2) +...+1)} −1 =2020×m −1 m=integer which is always 1 less than multiple of 2020 . Proved

$$\left(\mathrm{101}+\mathrm{21}\right)^{\mathrm{n}} \:−\left(\mathrm{101}+\mathrm{1}\right)^{\mathrm{n}} −\mathrm{21}^{\mathrm{n}} \\ $$$$=\left\{\:\mathrm{101}^{\mathrm{n}} +\mathrm{n101}^{\mathrm{n}−\mathrm{1}} \mathrm{21}+......+\mathrm{n101}.\left(\mathrm{21}\right)^{\mathrm{n}−\mathrm{1}} +\mathrm{21}^{\mathrm{n}} \right\}−\left\{\mathrm{101}^{\mathrm{n}} +\mathrm{n101}^{\mathrm{n}−\mathrm{1}} +.....+\mathrm{n101}+\mathrm{1}\right)−\mathrm{21}^{\mathrm{n}} \\ $$$$=\:\mathrm{n101}^{\mathrm{n}−\mathrm{1}} \left(\mathrm{21}−\mathrm{1}\right)+\:........+\mathrm{n101}\left(\mathrm{21}^{\mathrm{n}−\mathrm{1}} −\mathrm{1}\right)\:−\mathrm{1} \\ $$$$=\:\mathrm{101}×\mathrm{20}\left\{\mathrm{101}^{\mathrm{n}−\mathrm{2}} +.......\mathrm{n}\left(\mathrm{21}^{\mathrm{n}−\mathrm{2}} +\mathrm{21}^{\mathrm{n}−\mathrm{2}} +...+\mathrm{1}\right)\right\}\:−\mathrm{1} \\ $$$$=\mathrm{2020}×\mathrm{m}\:−\mathrm{1}\:\:\:\boldsymbol{\mathrm{m}}=\boldsymbol{\mathrm{integer}} \\ $$$$\boldsymbol{\mathrm{which}}\:\boldsymbol{\mathrm{is}}\:\boldsymbol{\mathrm{always}}\:\mathrm{1}\:\boldsymbol{\mathrm{less}}\:\boldsymbol{\mathrm{than}}\:\boldsymbol{\mathrm{multiple}}\:\boldsymbol{\mathrm{of}}\:\mathrm{2020}\:.\:\boldsymbol{\mathrm{Proved}} \\ $$

Commented by Tawa1 last updated on 07/Jul/19

God bless you sir

$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir} \\ $$

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