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

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

\dots \dots

\dots .

Answered by john santu last updated on 05/Jul/20

\Leftrightarrow x = 5 p + 2 \dots (\times 4)

\Leftrightarrow x = 4 q + 3 \dots (\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