if-between-10-4-and-10-n-there-are-9999000-coprime-numbers-with-20-find-n-please-thanks-

Question Number 213261 by lmcp1203 last updated on 02/Nov/24

i f b e t w e e n 10^{4} a n d 10^{n} t h e r e a r e 9999000 c o p r i m e n u m b e r s w i t h 20 . f i n d n . p l e a s e . t h a n k s

Answered by MrGaster last updated on 02/Nov/24

10^{4} < x < 10^{n}, (x, 20) = 1

20 = 2^{2} \times 5

ϕ (20) = 20 \times (1 - \frac{1}{2}) \times (1 - \frac{1}{5}) = 8

\frac{(10^{n} - 10^{4})}{20} \times ϕ (20) = 9999000

\frac{(10^{n} - 10^{4})}{20} \times 8 = 9999000

(10^{n} - 10^{4}) = \frac{9999000 \times 20}{8}

10^{n} - 10^{4} = 24997500

10^{n} = 25000000 + 10^{4}

10^{n} = 25010000

n = \log_{10} (2.501 \times 10^{7})

n = \log_{10} (2.501) + \log_{10} (10^{7})

n = \log_{10} (2.501) + 7

n \approx 0.398 + 7

n \approx 7.398

since n must be an integer,

\begin{matrix} n = 8 \end{matrix}

Commented by mr W last updated on 02/Nov/24

b u t b e t w e e n 10^{4} a n d 10^{8} t h e r e a r e

39 996 000 c o p r i m e s w i t h 20, n o t

9 999 000 .

Answered by lmcp1203 last updated on 02/Nov/24

t h a n k u o u .

Answered by mr W last updated on 02/Nov/24

20 = 2^{2} \times 5

a c o p r i m e w i t h 20 s h o u l d n o t c o n t a i n

t h e p r i m e f a c t o r 2 a n d 5 . t h a t m e a n s

a l l n u m b e r s b e t w e e n 10^{4} a n d 10^{n},

w h i c h a r e a m u l t i p l e o f 2 o r o f 5 o r

o f b o t h m u s t b e e l i m i n a t e d .

(10^{n} - 10^{4}) (1 - \frac{1}{2} - \frac{1}{5} + \frac{1}{10}) = 9999000

10^{n} = 25 007 500

t h i s i s n o t p o s s i b l e, s i n c e n i s n a t u r a l

n u m b e r .

t h a t m e a n s i t i s n o t p o s s i b l e t h a t

t h e r e a r e e x a c t l y 9999000 c o p r i m e s

w i t h 20 b e t w e e n 10^{4} a n d 10^{n}!

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