P-is-a-prime-number-P-gt-1000-If-P-r-mod-1000-how-many-value-of-r-

Question Number 203566 by BaliramKumar last updated on 22/Jan/24

^{'} P^{'} is a prime number (P > 1000) .

If P \equiv r (\mod 1000) . how many value of^{'} r^{'} .

Answered by MM42 last updated on 22/Jan/24

p = 1000 k + r & 0 < r < 1000

\Rightarrow “ p^{″} i s p r i m e n u m b e r

\Rightarrow “ r^{″} m u s t b e a n o d d n u m b e r & r \neq 5 k^{'}

A = {1, 3, 5, \dots, 999} \Rightarrow n (A) = 500

B = {5, 15, \dots, 995} \Rightarrow n (B) = 100

a n s = 500 - 100 = 400 ✓

f o r e x a m p l e

3001, 2003, 4007, \dots, 1993, 6997, 1999, . .

Commented by BaliramKumar last updated on 23/Jan/24

Can we use {Euler}^{'} s totient function ?

I {don}^{'} t know if {it}^{'} s true forever .

ϕ (1000) = ϕ [{(2 \times 5)}^{3}] = (2 - 1) (5 - 1) 2^{3 - 1} \times 5^{3 - 1} = 400

Commented by MM42 last updated on 23/Jan/24

I {d o n}^{'} t k h o w

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