Enter modulo statements Using the Chinese Remainder Theorem, solve:
x ≡ 2 mod 3
x ≡ 5 mod 7
x ≡ 1 mod 8
Pairwise Coprime: Take the GCF of 3 and modulus GCF(3,7) = 1
GCF(3,8) = 1
Pairwise Coprime: Take the GCF of 7 and modulus GCF(7,8) = 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 = 3 x 7 x 8
N = 168
Determine Equation Coefficients ci
Calculate c1 c1 = 56
Calculate c2 c2 = 24
Calculate c3 c3 = 21
Our equation becomes: x = a1 (c1 y1 ) + a2 (c2 y2 ) + a3 (c3 y3 )
x = a1 (56y1 ) + a2 (24y2 ) + a3 (21y3 )
Note: The ai piece is factored out
We will use this below
Calculate each y1 Using 1 modulus of 3 and c1 = 56 calculate y1 in the equation below:
3x
1 + 56y
1 = 1
y1 = -1
Calculate each y2 Using 2 modulus of 7 and c2 = 24 calculate y2 in the equation below:
7x
2 + 24y
2 = 1
y2 = -2
Calculate each y3 Using 3 modulus of 8 and c3 = 21 calculate y3 in the equation below:
8x
3 + 21y
3 = 1
y3 = -3
Plug in y values x = a1 (56y1 ) + a2 (24y2 ) + a3 (21y3 )
x = 2 x 56 x -1 + 5 x 24 x -2 + 1 x 21 x -3
x = -112 - 240 - 63
x = -415
Equation 1: Plug in -415 into modulus equations -415 ≡ 2 mod 3
3 x -139 = -417
Add remainder of 2 to -417 = -415
Equation 2: Plug in -415 into modulus equations -415 ≡ 5 mod 7
7 x -60 = -420
Add remainder of 5 to -420 = -415
Equation 3: Plug in -415 into modulus equations -415 ≡ 1 mod 8
8 x -52 = -416
Add remainder of 1 to -416 = -415
Final Answer -415
You have 2 free calculationss remaining
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?
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
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