Let-P-is-a-prime-number-P-gt-1000-If-P-devided-by-1000-then-remainder-is-r-How-many-value-of-r-

Question Number 193286 by BaliramKumar last updated on 09/Jun/23

Let ′P′ is a prime number (P > 1000). If ′P′ devided by 1000, then remainder is ′r′. How many value of ′r′ ?

$$\mathrm{Let}\:'\mathrm{P}'\:\mathrm{is}\:\mathrm{a}\:\mathrm{prime}\:\mathrm{number}\:\left(\mathrm{P}\:>\:\mathrm{1000}\right). \\ $$$$\mathrm{If}\:\:'\mathrm{P}'\:\mathrm{devided}\:\mathrm{by}\:\mathrm{1000},\:\mathrm{then}\:\mathrm{remainder}\:\mathrm{is}\:'\mathrm{r}'. \\ $$$$\mathrm{How}\:\mathrm{many}\:\mathrm{value}\:\mathrm{of}\:\:'\mathrm{r}'\:? \\ $$

Commented by Tinku Tara last updated on 09/Jun/23

All number other than multiple of 2 and 5 are possible.

$$\mathrm{All}\:\mathrm{number}\:\mathrm{other}\:\mathrm{than}\:\mathrm{multiple} \\ $$$$\mathrm{of}\:\mathrm{2}\:\mathrm{and}\:\mathrm{5}\:\mathrm{are}\:\mathrm{possible}. \\ $$

Commented by BaliramKumar last updated on 09/Jun/23

$$\mathrm{yes},\:\mathrm{but}\:\mathrm{total}\:\mathrm{value}\:\mathrm{of}\:\boldsymbol{\mathrm{r}}\:=\:? \\ $$

Answered by BaliramKumar last updated on 09/Jun/23

𝚽(1000) = 𝚽(10^3 ) = 4×10^(3−1) = 4×10^2 = 400

$$\boldsymbol{\Phi}\left(\mathrm{1000}\right)\:=\:\boldsymbol{\Phi}\left(\mathrm{10}^{\mathrm{3}} \right)\:=\:\:\mathrm{4}×\mathrm{10}^{\mathrm{3}−\mathrm{1}} \:=\:\mathrm{4}×\mathrm{10}^{\mathrm{2}} \:=\:\mathrm{400} \\ $$

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Let-P-is-a-prime-number-P-gt-1000-If-P-devided-by-1000-then-remainder-is-r-How-many-value-of-r-

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