use the Chinese Remainder theorem to find x such that x ≡ 2(mod 3) 2x ≡ 3(mod 5) 3x≡ 4( mod 7)


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Question Number 86240 by Rio Michael last updated on 27/Mar/20

$use the Chinese Remainder theorem to find$ $x such that$ $x \equiv 2 (\mod 3)$ $2 x \equiv 3 (\mod 5)$ $3 x \equiv 4 (\mod 7)$
	Answered by mr W last updated on 28/Mar/20

	$x = 3 k + 2$ $2 x = 6 k + 4 = 5 h + 3 \Rightarrow 5 h - 6 k = 1 . . . (i)$ $3 x = 9 k + 6 = 7 j + 4 \Rightarrow 7 j - 9 k = 2 . . . (i i)$ $(i) :$ $k = 5 n - 1, h = 6 n - 1$ $(i i) :$ $k = 7 m - 1, j = 9 m - 1$ $5 n - 1 = 7 m - 1 \Rightarrow 5 n - 7 m = 0 . . . (i i i)$ $(i i i) :$ $n = 7 p, m = 5 p$ $\Rightarrow k = 5 n - 1 = 35 p - 1$ $\Rightarrow x = 3 k + 2 = 105 p - 1 = 104, 209, 314, . . .$
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