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Let-N-be-a-natural-number-where-N-100-If-HCF-N-100-1-then-find-the-sum-of-all-the-values-of-N-a-400-b-1000-c-2000-d-4000-




Question Number 194528 by BaliramKumar last updated on 09/Jul/23
★ Let N be a natural number where N≤100.          If HCF(N, 100) = 1 then find the sum of           all the values of  N ?         (a) 400       (b) 1000        (c) 2000        (d) 4000
$$\bigstar\:\mathrm{Let}\:\mathrm{N}\:\mathrm{be}\:\mathrm{a}\:\mathrm{natural}\:\mathrm{number}\:\mathrm{where}\:\mathrm{N}\leq\mathrm{100}. \\ $$$$\:\:\:\:\:\:\:\:\mathrm{If}\:\mathrm{HCF}\left(\mathrm{N},\:\mathrm{100}\right)\:=\:\mathrm{1}\:\mathrm{then}\:\mathrm{find}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\: \\ $$$$\:\:\:\:\:\:\:\:\mathrm{all}\:\mathrm{the}\:\mathrm{values}\:\mathrm{of}\:\:\mathrm{N}\:? \\ $$$$\:\:\:\:\:\:\:\left(\mathrm{a}\right)\:\mathrm{400}\:\:\:\:\:\:\:\left(\mathrm{b}\right)\:\mathrm{1000}\:\:\:\:\:\:\:\:\left(\mathrm{c}\right)\:\mathrm{2000}\:\:\:\:\:\:\:\:\left(\mathrm{d}\right)\:\mathrm{4000} \\ $$
Answered by mahdipoor last updated on 09/Jul/23
(1+2+3+...+100)−(2+4+6+...100)  −(5+10+15+...+100)+(10+20+30+...+100)  =((100×101)/2)−((50×102)/2)−((20×105)/2)+((10×110)/2)  =2000
$$\left(\mathrm{1}+\mathrm{2}+\mathrm{3}+…+\mathrm{100}\right)−\left(\mathrm{2}+\mathrm{4}+\mathrm{6}+…\mathrm{100}\right) \\ $$$$−\left(\mathrm{5}+\mathrm{10}+\mathrm{15}+…+\mathrm{100}\right)+\left(\mathrm{10}+\mathrm{20}+\mathrm{30}+…+\mathrm{100}\right) \\ $$$$=\frac{\mathrm{100}×\mathrm{101}}{\mathrm{2}}−\frac{\mathrm{50}×\mathrm{102}}{\mathrm{2}}−\frac{\mathrm{20}×\mathrm{105}}{\mathrm{2}}+\frac{\mathrm{10}×\mathrm{110}}{\mathrm{2}} \\ $$$$=\mathrm{2000} \\ $$
Commented by BaliramKumar last updated on 09/Jul/23
typo sir 4000→2000  Nice solution
$$\mathrm{typo}\:\mathrm{sir}\:\mathrm{4000}\rightarrow\mathrm{2000} \\ $$$$\mathrm{Nice}\:\mathrm{solution} \\ $$
Commented by mahdipoor last updated on 09/Jul/23
you are right , i forget (/2) !
$${you}\:{are}\:{right}\:,\:{i}\:{forget}\:\frac{}{\mathrm{2}}\:! \\ $$
Answered by BaliramKumar last updated on 09/Jul/23
Apply Euler′s  totient function  Sum = 100×((𝛗(100))/2) = 100×((40)/2) = 100×20 = 2000
$$\mathrm{Apply}\:\mathrm{Euler}'\mathrm{s}\:\:\mathrm{totient}\:\mathrm{function} \\ $$$$\mathrm{Sum}\:=\:\mathrm{100}×\frac{\boldsymbol{\phi}\left(\mathrm{100}\right)}{\mathrm{2}}\:=\:\mathrm{100}×\frac{\mathrm{40}}{\mathrm{2}}\:=\:\mathrm{100}×\mathrm{20}\:=\:\mathrm{2000} \\ $$$$ \\ $$

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