Using the Chinese Remainder Theorem, solve:
x ≡ 1 mod 2
x ≡ 2 mod 3
x ≡ 3 mod 5
x ≡ 4 mod 11
Pairwise Coprime: Take the GCF of 2 and modulus
GCF(2,3) = 1
GCF(2,5) = 1
GCF(2,11) = 1
Pairwise Coprime: Take the GCF of 3 and modulus
GCF(3,5) = 1
GCF(3,11) = 1
Pairwise Coprime: Take the GCF of 5 and modulus
GCF(5,11) = 1
Coprime check
Since all 6 GCF calculations equal 1
the ni's are pairwise coprime
We can use the regular CRT Formula
Calculate the moduli product N
Take the product of each ni
N = n1 x n2 x n3 x n4
N = 2 x 3 x 5 x 11
N = 330
Determine Equation Coefficients ci
| ci = | N |
| ni |
Calculate c1
| c1 = | 330 |
| 2 |
c1 = 165
Calculate c2
| c2 = | 330 |
| 3 |
c2 = 110
Calculate c3
| c3 = | 330 |
| 5 |
c3 = 66
Calculate c4
| c4 = | 330 |
| 11 |
c4 = 30
Our equation becomes:
x = a1(c1y1) + a2(c2y2) + a3(c3y3) + a4(c4y4)
x = a1(165y1) + a2(110y2) + a3(66y3) + a4(30y4)
Note: The ai piece is factored out
We will use this below
Calculate each y1
Using 1 modulus of 2 and c1 = 165
calculate y1 in the equation below:
Calculate each y2
Using 2 modulus of 3 and c2 = 110
calculate y2 in the equation below:
Calculate each y3
Using 3 modulus of 5 and c3 = 66
calculate y3 in the equation below:
Calculate each y4
Using 4 modulus of 11 and c4 = 30
calculate y4 in the equation below:
Plug in y values
x = a1(165y1) + a2(110y2) + a3(66y3) + a4(30y4)
x = 1 x 165 x 1 + 2 x 110 x -1 + 3 x 66 x 1 + 4 x 30 x -4
x = 165 - 220 + 198 - 480
x = -337
Equation 1: Plug in -337 into modulus equations
-337 ≡ 1 mod 2
Add remainder of 1 to -338 = -337
Equation 2: Plug in -337 into modulus equations
-337 ≡ 2 mod 3
Add remainder of 2 to -339 = -337
Equation 3: Plug in -337 into modulus equations
-337 ≡ 3 mod 5
Add remainder of 3 to -340 = -337
Equation 4: Plug in -337 into modulus equations
-337 ≡ 4 mod 11
Add remainder of 4 to -341 = -337
Final Answer
What is the Answer?
How does the Chinese Remainder Theorem Calculator work?
x ≡ a mod b
x ≡ c mod d
x ≡ e mod f
the calculator will use the Chinese Remainder Theorem to find the lowest possible solution for x in each modulus equation.
Given that the ni portions are not pairwise coprime and you entered two modulo equations, then the calculator will attempt to solve using the Method of Successive Subsitution
This calculator has 1 input.
What 1 formula is used for the Chinese Remainder Theorem Calculator?
What 10 concepts are covered in the Chinese Remainder Theorem Calculator?
- algorithm
- A process to solve a problem in a set amount of time
- chinese remainder theorem
- ancient theorem that gives the conditions necessary for multiple equations to have a simultaneous integer solution
- coefficient
- a numerical or constant quantity placed before and multiplying the variable in an algebraic expression
- equation
- a statement declaring two mathematical expressions are equal
- gcf
- greatest common factor - largest positive integer dividing a set of integers
- modulus
- the remainder of a division, after one number is divided by another.
a mod b - product
- The answer when two or more values are multiplied together
- remainder
- The portion of a division operation leftover after dividing two integers
- substitution
- a simple way to solve linear equations algebraically and find the solutions of the variables.
- theorem
- A statement provable using logic
