Question on application of Chinese Remainder Theorem x <mspace width="thickmathspace" />

Semaj Christian 2022-06-26 Answered
Question on application of Chinese Remainder Theorem
x 3 ( mod 30 )
x 5 ( mod 56 )
I have a system of modular equation that I want to solve. However, I thought that this system has no solution because the modulos are not coprime. Further, attempting to solve using chinese remainder theorem:
x 56 p + 30 q
where p is such that
56 p 3 ( mod 30 )
and q is such that
30 q 5 ( mod 56 )
However, again the modular inverses of these do not exist.
Yet, one solution to this system of modular inequalities is 1293. How come the Chinese remainder theorem gives that no solution exists?
You can still ask an expert for help

Want to know more about Inequalities systems and graphs?

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

Govorei9b
Answered 2022-06-27 Author has 21 answers
Better if you write it as
x 3 ( mod 2 ) ,
x 3 ( mod 15 ) ,
x 5 ( mod 8 ) ,
x 5 ( mod 7 ) .
The ones with 2 and 8 are consistent, as the one with 2 is just asking for an odd number, so the system is equivalent to
x 3 ( mod 15 ) ,
x 5 ( mod 8 ) ,
x 5 ( mod 7 ) ,
where now the moduli are coprime. That is the requirement for the Chinese Remainder Theorem.
So we combine second and third again to get
x 3 ( mod 15 ) ,
x 5 ( mod 56 ) .
For the 56 one, we have
5 , 61 , 117 , 173 , 229 , 285 , 341 , 397 , 453 ,
and
453 3 ( mod 15 ) .
Did you like this example?
Subscribe for all access

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2022-06-04
Compute volume of E = { 5 x 8 y 7 x 8 ; 2 y 5 z 3 y 5 ; z 7 x 6 z 7 }
My progress:
It is easy to see that x , y , z 0. Therefore I can add all inequalities and after simple algebra get the following:
5 x 8 + 2 y 5 + z 7 x + y + z 7 x 8 + 3 y 5 + 6 x 7
asked 2022-05-14
Find the extrema of f subject to the stated constraints.
f ( x , y ) = x y, subject to x 2 y 2 = 2
I'm solving a problem involving Lagrange's multipliers, and I've got this system of equations to solve:
1 = λ 2 x
1 = λ 2 y
x 2 y 2 2 = 0
asked 2022-01-07
Use the given graph off to find a number δ such that if |x1|<δ then |f(x)1|<0.2
asked 2022-06-11
Notation of a linear inequality system.
x 1 0
x 2 0
x 3 0
is that right?
But what can I get from line 1? Thanks deeply for your help.
min { 2 x 1 + 4 x 2 + 7 x 3 : 2 x 1 + x 2 + 6 x 3 5 4 x 1 6 x 2 + 5 x 3 8 x 1 ,             x 2 ,           x 3 0 }
asked 2022-07-15
Find the intersection of two or more polygons in terms of linear inequalities
Given two or more closed polygons each defined by a system of linear inequalities, is there any method by which their intersection polygon may be determined, also in terms of a system of linear inequalities?
asked 2022-06-25
In an oil mill, they decide to make a mixture from two types of oil: the extra virgin whose price is $4 per liter and the virgin $3 per liter. They have 200 L of extra virgin olive oil and 500 L of virgin olive oil and they also want the cost of the mixed oil to be at least 1500 dollars.
The extra virgin olive oil will be e, while the virgin olive oil will be v.
a) Set up a system of inequalities that contains the restrictions set out in the problem.
The total number of mixed oil is:
e + v = m
200 + 500 = 700
Considering 4 e + 3 v, the maximum total price of the mixed oil is 2300:
4 ( 200 ) + 3 ( 500 ) = 2300
Then, I've formed a system of inequalities
{ e + v 700 4 e + 3 v 1500
I'm not sure if I've transformed the problem into a system correctly.
b) Is it possible to meet the conditions if the mixture contains the same amount of the two oils?
Since I wasn't sure about my system of equations, I've done the last two questions using
4 e + 3 v
I'm taking the max amount of extra virgin oil, 200L, and this is what we get:
4 ( 200 ) + 3 ( 200 ) = 800 + 600 = $ 1400
So, the answer would be
1400 < 1500
So, no, it would meet the conditions.
c) If they decide to use all the extra virgin oil they have in the mixture, what will be the amount of virgin oil that they should use if they want the cost of the mixture to be $ 1700?
1700 = 4 ( 200 ) + 3 v
1700 800 3 = v
300 L = v
If you could help me identify my errors I would greatly appreciate it.
asked 2022-05-20
Given: For some X, V a r ( X ) = 9, E ( X ) = 2, E ( X 2 ) = 13
Problem: P r [ X = 2 ] > 0
The solution in my book says to construct a r.v. to satisfy the above conditions and confirm or deny the statement from there. For simplicity, assume X can take on two values P r [ a ] = 1 2 and P r [ b ] = 1 2 . Finding that a and b are not 2 is enough to disprove the statement. We can apply the constraints:
1 2 a + 1 2 b = 2
1 2 a 2 + 1 2 b 2 = 13
So I did this:
a = 4 b from the first equation
b 2 4 b 5 = 0 by substituting a
The above quadratic has two solutions, b = 1 and b = 5 which happen to be the same solutions the book got for a and b respectively.
I'm not sure if this is the right way to solve this problem since I never explicitly solved for a and the two solutions for b may coincidentally be the same.