N-lt-10200-N-has-five-digits-N-22-23-and-N-5-17-Determinate-the-integer-N-

Question Number 125670 by mathocean1 last updated on 12/Dec/20

N < 10200, N h a s f i v e d i g i t s .

N \equiv 22 [23] a n d N \equiv 5 [17] .

D e t e r m i n a t e t h e i n t e g e r N .

Answered by floor(10²Eta[1]) last updated on 12/Dec/20

10000 ⩽ N < 10200

N \equiv 22 (\mod 23) \Rightarrow N = 23 a + 22, a \in Z

N \equiv 5 (\mod 17) \Rightarrow 23 a + 22 \equiv 6 a + 5 \equiv 5 (\mod 17)

\Rightarrow 6 a \equiv 0 (\mod 17) \Rightarrow a \equiv 0 (\mod 17)

\Rightarrow a = 17 b ∴ N = 391 b + 22, b \in Z

10000 ⩽ 391 b + 22 < 10200

9978 ⩽ 391 b < 10178

26 ⩽ b ⩽ 26 \Rightarrow b = 26

\Rightarrow N = 391.26 + 22

N = 10188