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Question Number 194528 by BaliramKumar last updated on 09/Jul/23

$$★\mathrm{Let}\mathrm{N}\mathrm{be}\mathrm{a}\mathrm{natural}\mathrm{number}\mathrm{where}\mathrm{N}\le 100.$$$$\mathrm{If}\mathrm{HCF}(\mathrm{N},100)=1\mathrm{then}\mathrm{find}\mathrm{the}\mathrm{sum}\mathrm{of}$$$$\mathrm{all}\mathrm{the}\mathrm{values}\mathrm{of}\mathrm{N}?$$$$\left(\mathrm{a}\right)400\left(\mathrm{b}\right)1000\left(\mathrm{c}\right)2000\left(\mathrm{d}\right)4000$$

Answered by mahdipoor last updated on 09/Jul/23

$$(1+2+3+...+100)-(2+4+6+...100)$$$$-(5+10+15+...+100)+(10+20+30+...+100)$$$$=\frac{100\times 101}{2}-\frac{50\times 102}{2}-\frac{20\times 105}{2}+\frac{10\times 110}{2}$$$$=2000$$

Commented by BaliramKumar last updated on 09/Jul/23

$$\mathrm{typo}\mathrm{sir}4000\to 2000$$$$\mathrm{Nice}\mathrm{solution}$$

Commented by mahdipoor last updated on 09/Jul/23

$$youareright,iforget\frac{}{2}!$$

Answered by BaliramKumar last updated on 09/Jul/23

$$\mathrm{Apply}{\mathrm{Euler}}^{\prime }\mathrm{s}\mathrm{totient}\mathrm{function}$$$$\mathrm{Sum}=100\times \frac{\mathit{\varphi }\left(100\right)}{2}=100\times \frac{40}{2}=100\times 20=2000$$$$$$

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