Introduction to the Chinese Remainder Theorem Calculator
Chinese Remainder Theorem Calculator with steps is an online tool that helps you to find the congruent values from the given equation with the help of the Chinese remainder theorem.
It is used to determine the unique solution up to a certain modulus of the given equation in a fraction of a second.
Chinese remainder calculator also simplifies your calculations with accuracy and easy by helping you solve equations involving modulus and the remainders.
What is the Chinese Remainder Theorem?
Chinese remainder theorem provides the unique value of modulus from the given system of congruence where coprime pairwise n1,n2,...nk and arbitrary integers a1,a2,...ak are given in the number system.
It has various applications in number theory, cryptography, computer science, and other fields of mathematics. It is used, for example, in the RSA encryption scheme and in solving various types of linear congruence systems.
Formula of Chinese Remainder Theorem
Chinese remainder theorem is based on a co-prime number and arbitrary integer that gives a unique solution of modulus. The Chinese remainder theorem calculator solves using the following equation,
$$ x \;≡\; a_1 \; (mod\; n_1) $$
$$ x \;≡\; a_2\; (mod\; n_2) $$
$$ .., x \;≡\; a_k \; (mod \;n_k) $$
Whereas,
a1,a2,a3,.... are co-prime numbers
n1,n2,n3,... are the integer of modulus
x= is the one value that satisfies the given congruent value
Use the Chinese Remainder Theorem Solver to quickly find solutions to congruence problems.
Working Method of the Chinese Remainder Calculator
Our tool uses the easiest method to solve the given equation with the help of Chinese remainder theorem principles. Its main reason is that our Chinese remainder theorem online calculator has advanced features to solve such a complex calculation quickly.
When you enter the input equation in the calculator. It starts the calculating process, in which it first finds out the common modulus N (N=n1*n2*..*nk). After finding the value of N, it uses the value of N and n1*n2*...*nk to find N1, N2,...Nn values. For this it divides the N/n1 for the N1 value, for N2=N/n2, and so on.
After calculation, we have N1, N2, N3,....Nn. Then it finds the inverse of N1, N2, N3,...Nn with respect to n1,n2,n3,..nk. Last, to get the value of x it multiply a1(arbitary coprime number) ,N1 and inverse of N1 as x=((a1*N1*N^-1)+(a2*N2*N^-2)+(a3*N3*N^-3)+.....+(ak*Nn*N^-n)).
After solving the value of x we get the unique solution of the modulus of the given equation. Let's see an example of the Chinese remainder theorem to see the working process behind our Chinese remainder calculator.
Solved Example of Chinese Remainder Theorem
An example of the Chinese remainder theorem with a solution is given to know how the Chinese remainder theorem calculator provides the step-by-step solution.
Example:
Find the system of congruence using the Chinese remainder theorem.
$$ x \;≡\; 2 \; (mod\;3) $$
$$ x \;≡\; 3\; (mod\; 5) $$
$$ x \;≡\; 2\; (mod\; 7) $$
Solution:
We have the given pairwise coprime number 3,5,7.
Now find the value of N where all the coprime are multiplied.
$$ N \;=\; 3 \times 5 \times 7 \;=\; 105 $$
Find the value of N1,N2,N3,...Nn as N1=N/n1 , N2=N/n2, N3=N/n3
$$ N_1 \;=\; \frac{105}{3} \;=\; 35 $$
$$ N_2 \;=\; \frac{105}{3} \;=\; 21 $$
$$ N_3 \;=\; \frac{105}{7} \;=\; 15 $$
Now find the inverse of N1, N2, N3,...Nn with respect to n1,n2,n3,..nk
$$ For\; N_1 \;=\; 35 \; abd n_1 \;=\; 3:35 \times 2 \;≡\; 1 \;(mod\; 3) $$
$$ For\;N_2 \;=\; 21\;and \; n_2 \;=\; 5:21 \times 1 \;≡\; 1\; (mod\; 5) $$
$$ For\; N_3 \;=\; 15 \;and\; n_3 \;=\; 7:15 \times 1 \;≡\; 1\; (mod\; 7) $$
Lastly for the value of x use this formula
$$ x \;=\; \biggr((a_1 \times N_1 \times N^{-1})+(a_2 \times N_2 \times N^{-2}) + (a_3 \times N_3 \times N^{-3})+.....+(ak \times N_n \times N^{-n}) \biggr) $$
$$ x \;=\; a_1 \times N_1 \times x_1 + a_2 \times N_2 \times x_2 + a_3 \times N_3 \times x_3 $$
$$ x \;=\; 2 \times 35 \times 2 + 3 \times 21 \times 1 + 2 \times 15 \times 1 $$
$$ x \;=\; 140 + 63 + 30 $$
$$ x \;=\; 233 $$
For the smallest value of x for nonnegative modulus value
$$ x \;≡\; 233\; (mod\; 105) $$
$$ x \;≡\; 23\; (mod\; 105) $$
For the system of congruence solution is x ☰ 23 (mod 105)
Instruction: Using the Chinese Remainder Theorem Calculator
Chinese remainder calculator has a simple interface so that you can use it to calculate the unique value of x in congruence value problems.
Before giving the input value in our Chinese remainder theorem solver, you must follow some instructions so that you do not have any inconvenience in the evaluation process. These steps are:
- Choose the number of congruent equations from the given list.
- Enter the system of congruence equation in the input field.
- Review your input problem because it is not true as it does not give you the solution of modulus of x in the Chinese remainder theorem.
- Click the “Calculate” button to get the result of the unique modulus value of x from the given system.
- Click on the “Recalculate” button to get a new page for solving more problems for Chinese remainder theorem.
Output of the Chinese Remainder Theorem Solver
Chinese remainder theorem calculator gives you the solution of the Eludicean remainder division value when you add the input to it. It provides you with solutions in a step-wise process. It may contain as:
- The result option gives you a solution for Chinese remainder theorem problems.
- The possible step provides you with all the steps of the problem of the Eludicean remainder division value of a given congruent value.
Benefits of Using the Chinese Remainder Theorem Online Calculator
Chinese remainder calculator will give you tons of advantages whenever you use it to calculate the unique remainder value of a given equation. These benefits are:
- lt saves time and effort from doing complex computations problems to get the unique solution of modulus.
- It is a free-of-cost tool so you can use it for free to find the value of the remainder value for the division problems in number theory.
- Chinese remainder theorem calculator with steps is a versatile tool that allows you to solve different types of complicated questions of Chinese remainder theorem problems.
- You can use this calculator for practice so that you may be familiar with the concept of the unique solution of modulus in an equation using the Chinese remainder theorem online calculator
- It is a trustworthy tool that provides you with accurate solutions every time whenever you use it to find the Chinese remainder theorem for congruent value equation calculation.
