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Hi-veterans-Serlea-1-Find-the-last-three-digits-of-3005-11-3005-12-3005-13-3005-3002-




Question Number 85183 by Serlea last updated on 19/Mar/20
Hi veterans  Serlea (1)    Find the last three digits of:  3005^(11) +3005^(12) +3005^(13) +...+3005^(3002)
$$\mathrm{Hi}\:\mathrm{veterans} \\ $$$$\mathrm{Serlea}\:\left(\mathrm{1}\right) \\ $$$$ \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{last}\:\mathrm{three}\:\mathrm{digits}\:\mathrm{of}: \\ $$$$\mathrm{3005}^{\mathrm{11}} +\mathrm{3005}^{\mathrm{12}} +\mathrm{3005}^{\mathrm{13}} +…+\mathrm{3005}^{\mathrm{3002}} \\ $$$$ \\ $$$$ \\ $$
Commented by mr W last updated on 19/Mar/20
let say A=^(3) B means “the last three  digits from A are equal to the last  three digits from B”.  3005^n =^(3) 5^n   5^(odd) =^(3) 125  (with odd≥3)  5^(even) =^(3) 625 (with even≥4)    3005^(11) +3005^(12) +3005^(13) +...+3005^(3002)   =^3 5^(11) +5^(12) +5^(13) +5^(14) +...+5^(3001) +5^(3002)   =^3 (125+625)+(125+625)+...+(125+625)  =^3 750×1496  =^3 000 ⇒answer
$${let}\:{say}\:{A}\overset{\mathrm{3}} {=}{B}\:{means}\:“{the}\:{last}\:{three} \\ $$$${digits}\:{from}\:{A}\:{are}\:{equal}\:{to}\:{the}\:{last} \\ $$$${three}\:{digits}\:{from}\:{B}''. \\ $$$$\mathrm{3005}^{{n}} \overset{\mathrm{3}} {=}\mathrm{5}^{{n}} \\ $$$$\mathrm{5}^{{odd}} \overset{\mathrm{3}} {=}\mathrm{125}\:\:\left({with}\:{odd}\geqslant\mathrm{3}\right) \\ $$$$\mathrm{5}^{{even}} \overset{\mathrm{3}} {=}\mathrm{625}\:\left({with}\:{even}\geqslant\mathrm{4}\right) \\ $$$$ \\ $$$$\mathrm{3005}^{\mathrm{11}} +\mathrm{3005}^{\mathrm{12}} +\mathrm{3005}^{\mathrm{13}} +…+\mathrm{3005}^{\mathrm{3002}} \\ $$$$\overset{\mathrm{3}} {=}\mathrm{5}^{\mathrm{11}} +\mathrm{5}^{\mathrm{12}} +\mathrm{5}^{\mathrm{13}} +\mathrm{5}^{\mathrm{14}} +…+\mathrm{5}^{\mathrm{3001}} +\mathrm{5}^{\mathrm{3002}} \\ $$$$\overset{\mathrm{3}} {=}\left(\mathrm{125}+\mathrm{625}\right)+\left(\mathrm{125}+\mathrm{625}\right)+…+\left(\mathrm{125}+\mathrm{625}\right) \\ $$$$\overset{\mathrm{3}} {=}\mathrm{750}×\mathrm{1496} \\ $$$$\overset{\mathrm{3}} {=}\mathrm{000}\:\Rightarrow{answer} \\ $$

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