Using the Chinese Remainder Theorem, solve the following system of mod

Using the Chinese Remainder Theorem, solve the following system of mod MathCelebrity logo Image to Crop Enter modulo statements

Using the Chinese Remainder Theorem, solve:

x ≡ 1 mod 5

x ≡ 2 mod 6

x ≡ 3 mod 7


Pairwise Coprime: Take the GCF of 5 and modulus

GCF(5,6) = 1

GCF(5,7) = 1


Pairwise Coprime: Take the GCF of 6 and modulus

GCF(6,7) = 1


Coprime check

Since all 3 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

N = 5 x 6 x 7

N = 210

Determine Equation Coefficients ci
ci  =  N
  ni

Calculate c1
c1  =  210
  5

c1 = 42

Calculate c2
c2  =  210
  6

c2 = 35

Calculate c3
c3  =  210
  7

c3 = 30

Our equation becomes:

x = a1(c1y1) + a2(c2y2) + a3(c3y3)

x = a1(42y1) + a2(35y2) + a3(30y3)

Note: The ai piece is factored out

We will use this below

Calculate each y1

Using 1 modulus of 5 and c1 = 42
calculate y1 in the equation below:

5x1 + 42y1 = 1

y1 = -2

Calculate each y2

Using 2 modulus of 6 and c2 = 35
calculate y2 in the equation below:

6x2 + 35y2 = 1

y2 = -1

Calculate each y3

Using 3 modulus of 7 and c3 = 30
calculate y3 in the equation below:

7x3 + 30y3 = 1

y3 = -3

Plug in y values

x = a1(42y1) + a2(35y2) + a3(30y3)

x = 1 x 42 x -2 + 2 x 35 x -1 + 3 x 30 x -3

x = -84 - 70 - 270

x = -424

Equation 1: Plug in -424 into modulus equations

-424 ≡ 1 mod 5

5 x -85 = -425

Add remainder of 1 to -425 = -424

Equation 2: Plug in -424 into modulus equations

-424 ≡ 2 mod 6

6 x -71 = -426

Add remainder of 2 to -426 = -424

Equation 3: Plug in -424 into modulus equations

-424 ≡ 3 mod 7

7 x -61 = -427

Add remainder of 3 to -427 = -424

Final Answer

-424

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What is the Answer?

How does the Chinese Remainder Theorem Calculator work?

Free Chinese Remainder Theorem Calculator - Given a set of modulo equations in the form:
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?

algorithmA process to solve a problem in a set amount of timechinese remainder theoremancient theorem that gives the conditions necessary for multiple equations to have a simultaneous integer solutioncoefficienta numerical or constant quantity placed before and multiplying the variable in an algebraic expressionequationa statement declaring two mathematical expressions are equalgcfgreatest common factor - largest positive integer dividing a set of integersmodulusthe remainder of a division, after one number is divided by another.
a mod bproductThe answer when two or more values are multiplied togetherremainderThe portion of a division operation leftover after dividing two integerssubstitutiona simple way to solve linear equations algebraically and find the solutions of the variables.theoremA statement provable using logic

Example calculations for the Chinese Remainder Theorem Calculator

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