N<10200 , N has five digits. N≡22[23] and N≡5[17]. Determinate the integer N.


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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

	$10000 ⩽ N < 10200$ $N \equiv 22 (\mod 23) \Rightarrow N = 23 a + 22, a \in Z$ $N \equiv 5 (\mod 17) \Rightarrow 23 a + 22 \equiv 6 a + 5 \equiv 5 (\mod 17)$ $\Rightarrow 6 a \equiv 0 (\mod 17) \Rightarrow a \equiv 0 (\mod 17)$ $\Rightarrow a = 17 b ∴ N = 391 b + 22, b \in Z$ $10000 ⩽ 391 b + 22 < 10200$ $9978 ⩽ 391 b < 10178$ $26 ⩽ b ⩽ 26 \Rightarrow b = 26$ $\Rightarrow N = 391.26 + 22$ $N = 10188$
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