If x an integer when divided 5 give remainder 2 and when divided 4 give remainder 3. find the value of x


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Question Number 101888 by bemath last updated on 05/Jul/20

$I f x a n i n t e g e r w h e n d i v i d e d$ $5 g i v e r e m a i n d e r 2 a n d w h e n d i v i d e d$ $4 g i v e r e m a i n d e r 3 .$ $f i n d t h e v a l u e o f x$
Commented by Rasheed.Sindhi last updated on 05/Jul/20

$x \equiv 2 (m o d 5)$ $x \equiv 3 (m o d 4)$ $U s e C h i n e s e r e m a i n d e r t h e o r m .$
	Answered by Rasheed.Sindhi last updated on 05/Jul/20

	$x - 2 i s d i v i s i b l e b y 5$ $x = 5 m + 2$ $x - 3 i s d i v i s i b l e b y 4$ $x = 4 n + 3$ $5 m + 2 = 4 n + 3$ $m = \frac{4 n + 1}{5}; m, n \in N$ $n = 1 \Rightarrow m = 1 \Rightarrow x = 7$ $n = 6 \Rightarrow m = 5 \Rightarrow x = 27$ $n = 11 \Rightarrow m = 9 \Rightarrow x = 47$ $. . . . . .$ $. . . .$
	Answered by john santu last updated on 05/Jul/20

	$\Leftrightarrow x = 5 p + 2 . . . (\times 4)$ $\Leftrightarrow x = 4 q + 3 . . . (\times 5)$ $_________-$ $- x = 20 (p - q) - 7$ $x = 20 (q - p) + 7$ $s o q - p \in Z$ $(1) q - p = 1 \Rightarrow x = 27$ $(2) q - p = 2 \Rightarrow x = 47$ $(3) q - p = 3 \Rightarrow x = 67$ $(4) q - p = 4 \Rightarrow x = 87$ $(5) q - p = 0 \Rightarrow x = 7$ $s o o n$
	Answered by mathmax by abdo last updated on 05/Jul/20

	$x \equiv 2 [5] and x \equiv 3 4] \Rightarrow x = 5 n + 2 and x = 4 m + 3 \Rightarrow$ $5 n + 2 = 4 m + 3 let use congruence modulo 5 (prime) \Rightarrow$ $\bar{2} = - \bar{m} + \bar{3} \Rightarrow \bar{m} = \bar{3} - \bar{2} = \bar{1} \Rightarrow m = 5 q + 1 \Rightarrow x = 4 (5 q + 1) + 3$ $\Rightarrow x = 20 q + 7$
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