Question

if 3x = 2(mod 7). find x

Answer 1

try, x = 1, 2, 3, 4, 5, 6

Answer 2

put x = 1

3*1 = 2 (mod 7)

FALSE

Answer 3

put x = 2

3*2 = 2 (mod 7)

FALSE

Answer 4

put x = 3

3*3 = 2 (mod 7)

TRUE.

Answer 5

so, x = 3

Answer 6

you can always try like this, when the mod value is less

Answer 7

actually whats mean 2(mod7) ?

Answer 8

the remainder of some number when divided by 7

Answer 9

2(mod 7)

means, when divided by 7, the remainder is 2

Answer 10

@ganeshie8 Is there someway in general?

Answer 11

yup !

Answer 12

x = 2 (mod 7)

means

7 | x - 2

Answer 13

by trying like that, we get just 1 solution, in fact, there are infinite

Answer 14

nope, there is oly one solution mod 7, since gcd(3, 7) = 1

Answer 15

3x = 2 (mod 7)

solution is :-

x = 3 (mod 7)

Answer 16

just discuss:
as you say x = 3 (mod7) means x -3 = 7n , so, if the right hand side is 21, so, x = 18
if the right hand side is 35, so, x = 33........ so???

Answer 17

thats right ! and all satisfy x = 3 (mod 7)

so we say there is a unique solution in mod7 world :)

Answer 18

I never took this course before, just read the book. That's why I am not sure about what I am saying , hehehe. thanks for explanation

Answer 19

np :) number theory is awesome. since im able to digest it, its not a big deal for u.. :)

Answer 20

thanks

Answer 21

thanks @ganeshie8 ,, but im still confuse about this,, and I want to ask another question about this,, maybe after that, i can understand,,
1) X=5(mod 7)
2)X=2(mod 5)

Answer 22

good :)
whats the question ?

Answer 23

let me give u a question :-

4x = 3 (mod 11)

Answer 24

find x

Answer 25

okay go on,, and i already mention the question just now,,

Answer 26

1) X=5(mod 7)
2)X=2(mod 5)

these are not questions :) these are answers ok :)

Answer 27

question is this :-
4x = 3 (mod 11)

find x

Answer 28

4x = 3 (mod 11)

what this guy actually asking is this :-

find x such that

4x -3 is divisible by 11

Answer 29

can u find SOME x such that, 4x-3 is divisible by 11 ha ?

Answer 30

ermmmm,,, hehehe,, dont know

Answer 31

tttrrrrrryyy :)

Answer 32

x = 5 (mod7)
x =2 (mod5)
solve for x , right? and they are a system, not separate, right?

Answer 33

yes, if thats one problem, then they are system. we can solve them using Chinese Remainder Theorem

Answer 34

yup, got you @ganeshie8 and I got the answer is x = 13 ( mod 35) hihihi

Answer 35

oww ! but im getting 12 (mod 35) :-

x = 5 (mod 7)
=>
x - 5 = 7k
x = 5+ 7k


x = 2 (mod 5)

5+7k = 2(mod 5)

k = 1 (mod 5)
k = 1 + 5m


so,

x = 5 + 7(1 + 5m)
= 5 + 7 + 35m
= 12 + 35m

Answer 36

@nadia.a do you want me guide you to understand how to solve his problem? 4x = 3 ( mod 11)?

Answer 37

yessss

Answer 38

@ganeshie8 I will show you mine after guiding the asker, we must meet at that problem because I apply chinese theorem.

Answer 39

@nadia.a in Z11 , the remainder of a division must < 11, right?

Answer 40

sure :) chinese and commonsense both should give same answer :)

Answer 41

so, the remainder should be one of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ,

Answer 42

the number is 4*x and we find out the remainder in Z11 is 3, now test
4*1 =4 and 11|4 = 0 remain 4 , got this or not?

Answer 43

yes i got it,,

Answer 44

4*2 =8 and 11|8 = 0 remain 8
4*3 = 12 and 11|12 = 1 remain1
4*4 =16 and 11|16 = 1 remain 5
4*5 = 20 and 11|20 = 1 remain 9
4*6 = 24 and 11|24 = 2 remain 2
4*7 =28 and 11|28 = 2 remain 6
4*8 = 32 and 11| 32 = 2 remain 10
4*9 = 36 and 11|36 = 3 remain 3
bingo, we got remain 3 hehehe. so 4*9 =3 (mod11) compare to 4*x = 3 (mod11)
we have 9 = x
got me?

Answer 45

Loser you're very fast learner xD

Answer 46

Thank you!!!

Answer 47

wish my brain works like u when i touch new subjects -_-

Answer 48

@ganeshie8 it's not new to me, just because I learned by reading the book, I didn't have the confirmation from the prof. When seeing it, I would like to practice under the observance of an expert like you

Answer 49

ahh thats kind of u, but i get what u mean :)

Answer 50

owhhhhhhhhhhhhhhh,, i see,, hahaha,, thanks both of u :D

Answer 51

ok, back to our problem, To Chinese theorem
x =5 (mod 7)
x =2 (mod 5) we have a = 2, b =4, m = 5 and n =7
so, x = a[n*n]+ b [m*1](mod m*n)
x = 2[7*7] + 4 [5*1] (mod 35)
x = 2 *49 + 20 (mod 35)
x = 118 (mod 35)
x = 13 (mod 35)
@ganeshie8

Answer 52

okay, before looking at where it went wrong wid chinese,
lets verify if the answer is correct or not :)

Answer 53

hypothesis :-
x =5 (mod 7)
x =2 (mod 5)

Answer :-
x = 13 (mod 35)

Answer 54

oh, yea, that 's the easiest way to confirm, ha!!

Answer 55

we ask two questions now for verifying the answer :-
1) Does 13 | 7 leave a remainder of of 5 ?
2) Does 13 | 5 leave a remainder of 2 ?

Answer 56

yup, I am wrong, hihihih... please, point it out.

Answer 57

you want the chinese way ha ?

Answer 58

aha

Answer 59

gimme 5 mnts to pull up chinese formula :)

Answer 60

@ganeshie8 I know what's wrong with me, b = 5 not 4

Answer 61

ack.... just a typo lol

Answer 62

Thaaaaaaaaaaaaanks a ton

Answer 63

x =5 (mod 7)
x =2 (mod 5)

solution = \(a_1 N_1x_1 + .... a_r N_r x_r (\mod 7 \times 5)\)

Answer 64

if we add one more such as X=3(mod 5)

Answer 65

you want to solve the system :-
x =5 (mod 7)
x =2 (mod 5)
x=3(mod 5)

?

Answer 66

the simple answer is NO solution,
cuz you cannot have below simultaneously :-
x =2 (mod 5)
x=3(mod 5)

Answer 67

a little change x = 3 (mod 11)

Answer 68

x =5 (mod 7)
x =2 (mod 5)
x=3(mod 11)

Answer 69

you wanto use chinese / commonsense ? :)

Answer 70

why NO ????

Answer 71

when you ask :-
x =2 (mod 5)
x=3(mod 5)

what you're asking is : gimme a number that leaves remainders 2 and 3 when divided by 5

Answer 72

there is no single number wid that property. so NO solution.

Answer 73

owh,, okay2,, i just noticed it :DD

Answer 74

good :)

Answer 75

you're also familiar wid chinese nadia ?

Answer 76

x =5 (mod 7)
x =2 (mod 5)
x=3(mod 11)

solution :\( 5 N_1x_1 + 2 N_2 x_2 + 3 N_3 x_3 (\mod 7 \times 5 \times 11) \)

Answer 77

\(N_1 = 5\times 11\)
\(N_2 = 7\times 11\)
\(N_3 = 7\times 5\)


\(N_1x_1 \equiv 1 (\mod 7)\)
\(N_2 x_2 \equiv 1 (\mod 5)\)
\(N_3x_3 \equiv 1 (\mod 11)\)

Answer 78

plug and chug. dont wry if they dint cover chinese yet, it wil come soon...

Answer 79

@ganeshie8 i want to ask,, when i review back ur question 4X=3(mod 11),, i understand now,, so if i got that kind of question in exam, i have to try one by one the integer to find the remainder ?

Answer 80

thats one way,
there is a standard way for solving linear congruences like this :-

4x = 3 (mod 11)

=>

4x -11y = 3
this is just a diaphontine equation.
familiar wid solving diaphontine equations ? :)

Answer 81

erm,, not really,, how ? tomorrow i hv exam about this topic,, help me :)

Answer 82

@ganeshie8