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

Enter modulo statements

Using the Chinese Remainder Theorem, solve:

x ≡ 1 mod 2

x ≡ 2 mod 3

x ≡ 3 mod 5

x ≡ 4 mod 11

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:

2x1 + 165y1 = 1

y1 = 1

Calculate each y2

Using 2 modulus of 3 and c2 = 110
calculate y2 in the equation below:

3x2 + 110y2 = 1

y2 = -1

Calculate each y3

Using 3 modulus of 5 and c3 = 66
calculate y3 in the equation below:

5x3 + 66y3 = 1

y3 = 1

Calculate each y4

Using 4 modulus of 11 and c4 = 30
calculate y4 in the equation below:

11x4 + 30y4 = 1

y4 = -4

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

2 x -169 = -338

Add remainder of 1 to -338 = -337

Equation 2: Plug in -337 into modulus equations

-337 ≡ 2 mod 3

3 x -113 = -339

Add remainder of 2 to -339 = -337

Equation 3: Plug in -337 into modulus equations

-337 ≡ 3 mod 5

5 x -68 = -340

Add remainder of 3 to -340 = -337

Equation 4: Plug in -337 into modulus equations

-337 ≡ 4 mod 11

11 x -31 = -341

Add remainder of 4 to -341 = -337

Final Answer

-337

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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

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

Pairwise Coprime: Take the GCF of 2 and modulus

Pairwise Coprime: Take the GCF of 3 and modulus

Pairwise Coprime: Take the GCF of 5 and modulus