Chinese Remainder Theorem Calculator
Visual Representation of Moduli and Remainders
This chart visually compares the magnitudes of your input moduli (nᵢ) and their respective remainders (aᵢ).
Welcome to the advanced CRT Calculator, your essential tool for solving systems of linear congruences using the Chinese Remainder Theorem. Whether you're a student, mathematician, or computer scientist, this calculator simplifies complex modular arithmetic problems. Input your congruences in the form x ≡ a (mod n) and let our calculator find the smallest non-negative integer solution.
This chart visually compares the magnitudes of your input moduli (nᵢ) and their respective remainders (aᵢ).
A CRT calculator is an online tool designed to solve systems of linear congruences based on the Chinese Remainder Theorem (CRT). This powerful theorem from number theory provides a unique solution (modulo the product of the moduli) to a system of congruences, provided certain conditions are met.
In essence, if you have multiple equations like:
x ≡ a₁ (mod n₁)x ≡ a₂ (mod n₂)x ≡ aₖ (mod nₖ)The CRT calculator helps you find an integer x that satisfies all these conditions simultaneously. This is particularly useful when the moduli (nᵢ) are pairwise coprime, meaning that any two moduli share no common factors other than 1.
The most frequent misunderstanding when using a CRT calculator or applying the Chinese Remainder Theorem is related to the moduli. For the standard CRT, the moduli (nᵢ) must be pairwise coprime. If they are not, a solution might not exist, or it might not be unique modulo the product of the moduli, requiring a more generalized approach. This calculator specifically focuses on the standard CRT where moduli are assumed coprime. Another point of confusion is that a and n values are unitless integers; they do not represent physical quantities or units like meters, seconds, or dollars.
The Chinese Remainder Theorem provides a method to find a unique solution x modulo N, given a system of congruences:
x ≡ a₁ (mod n₁)
x ≡ a₂ (mod n₂)
...
x ≡ aₖ (mod nₖ)
where n₁, n₂, ..., nₖ are pairwise coprime positive integers, and aᵢ are integers.
The solution x is given by:
x ≡ (a₁N₁y₁ + a₂N₂y₂ + ... + aₖNₖyₖ) (mod N)
Where:
N: The product of all moduli: N = n₁ * n₂ * ... * nₖ.Nᵢ: For each congruence i, Nᵢ = N / nᵢ.yᵢ: The modular multiplicative inverse of Nᵢ modulo nᵢ. That is, yᵢ satisfies Nᵢyᵢ ≡ 1 (mod nᵢ). This inverse exists because nᵢ and Nᵢ are coprime (since all n values are pairwise coprime).| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
aᵢ |
Remainder for congruence i |
Unitless Integer | 0 to nᵢ - 1 |
nᵢ |
Modulus for congruence i |
Unitless Integer | > 1 |
N |
Product of all moduli (n₁ * ... * nₖ) |
Unitless Integer | Can be very large |
Nᵢ |
Product of all moduli except nᵢ (N / nᵢ) |
Unitless Integer | Can be very large |
yᵢ |
Modular inverse of Nᵢ modulo nᵢ |
Unitless Integer | 0 to nᵢ - 1 |
x |
The solution, the smallest non-negative integer satisfying all congruences | Unitless Integer | 0 to N - 1 |
A farmer is counting eggs. When she puts them in groups of 3, there are 2 eggs left. When she puts them in groups of 5, there are 3 eggs left. When she puts them in groups of 7, there are 2 eggs left. How many eggs does she have, assuming she has the smallest possible number?
x ≡ 2 (mod 3)x ≡ 3 (mod 5)x ≡ 2 (mod 7)Using the CRT calculator, we find:
N = 3 * 5 * 7 = 105N₁ = 105 / 3 = 35; y₁ = 35⁻¹ (mod 3) = 2 (since 35 ≡ 2 mod 3, and 2*2 = 4 ≡ 1 mod 3)N₂ = 105 / 5 = 21; y₂ = 21⁻¹ (mod 5) = 1 (since 21 ≡ 1 mod 5, and 1*1 = 1 ≡ 1 mod 5)N₃ = 105 / 7 = 15; y₃ = 15⁻¹ (mod 7) = 1 (since 15 ≡ 1 mod 7, and 1*1 = 1 ≡ 1 mod 7)x ≡ (2*35*2 + 3*21*1 + 2*15*1) (mod 105)x ≡ (140 + 63 + 30) (mod 105)x ≡ 233 (mod 105)x ≡ 23 (mod 105)The smallest number of eggs is 23.
In a simplified cryptographic scenario, suppose a secret key x needs to satisfy these conditions:
x ≡ 1 (mod 4)x ≡ 4 (mod 9)x ≡ 11 (mod 13)Find the smallest positive integer x.
Using the CRT calculator:
N = 4 * 9 * 13 = 468N₁ = 468 / 4 = 117; y₁ = 117⁻¹ (mod 4) = 1 (since 117 ≡ 1 mod 4, and 1*1 = 1 ≡ 1 mod 4)N₂ = 468 / 9 = 52; y₂ = 52⁻¹ (mod 9) = 4 (since 52 ≡ 7 mod 9, and 7*4 = 28 ≡ 1 mod 9)N₃ = 468 / 13 = 36; y₃ = 36⁻¹ (mod 13) = 4 (since 36 ≡ 10 mod 13, and 10*4 = 40 ≡ 1 mod 13)x ≡ (1*117*1 + 4*52*4 + 11*36*4) (mod 468)x ≡ (117 + 832 + 1584) (mod 468)x ≡ 2533 (mod 468)x ≡ 193 (mod 468) (since 2533 = 5 * 468 + 193)The smallest secret key x is 193.
Note: The manual calculations above demonstrate the process. The calculator will perform these steps accurately. It's crucial to double-check modular inverses.
Our CRT calculator is designed for ease of use, allowing you to quickly find solutions to your Chinese Remainder Theorem problems. Follow these simple steps:
x ≡ a (mod n), enter the remainder a in the "Remainder (a)" field and the modulus n in the "Modulus (n)" field.
a is a non-negative integer less than n.n is a positive integer greater than 1.x that satisfies all your input congruences.x + kN).N, the sum of terms, and a table showing Nᵢ, yᵢ, and aᵢNᵢyᵢ for each congruence. This helps in understanding the theorem's application.The solution derived from the Chinese Remainder Theorem, and its applicability, are influenced by several critical factors:
nᵢ): This is the most crucial factor for the standard CRT. If the moduli are not pairwise coprime, a solution might not exist, or it might not be unique modulo the product of the moduli. Our CRT calculator will warn you if this condition is not met, though it attempts to find a solution even then (which might not be the "standard" unique one).nᵢ): Larger moduli lead to a larger product N, which in turn means the solution x (modulo N) can be a much larger number. This affects the range of the unique solution.N and thus the complexity of the problem and the potential magnitude of x. Each additional congruence adds another constraint that x must satisfy.aᵢ): The specific remainders dictate the final value of x. Changing even one aᵢ can drastically alter the solution.yᵢ): The formula relies on finding yᵢ = Nᵢ⁻¹ (mod nᵢ). This inverse is guaranteed to exist if gcd(Nᵢ, nᵢ) = 1. Since Nᵢ is the product of all other coprime moduli, and nᵢ is coprime to them, gcd(Nᵢ, nᵢ) = 1 is always true in the standard CRT.A: The Chinese Remainder Theorem (CRT) is a powerful concept in number theory that provides a way to find a unique solution to a system of linear congruences. It states that if you have several congruences, each with a different modulus, and these moduli are pairwise coprime, then there is a unique solution modulo the product of all the moduli.
A: The pairwise coprimality ensures that a unique solution exists modulo the product of the moduli. It also guarantees the existence of the modular inverses required in the formula. If moduli share common factors, the system might have no solution, or a solution that is unique only modulo the least common multiple (LCM) of the moduli, requiring a more generalized approach.
A: This CRT calculator is primarily designed for the standard CRT where moduli are pairwise coprime. If they are not, it will issue a warning. While some generalized CRT algorithms exist for non-coprime moduli, this calculator might not yield the expected unique solution or might indicate no solution if the congruences are inconsistent. It's best to ensure your moduli are coprime for accurate results with this tool.
A: Yes, the solution x found by the CRT is unique modulo N (the product of all moduli). This means there are infinitely many solutions, but they all differ by multiples of N. The calculator provides the smallest non-negative integer solution.
A: A modular inverse of a number a modulo n is a number y such that (a * y) ≡ 1 (mod n). It's essentially the modular equivalent of division. In the CRT formula, modular inverses (yᵢ) are used to "undo" the effect of multiplication by Nᵢ when working modulo nᵢ, allowing us to combine the individual congruence solutions.
A: Absolutely! The CRT has significant applications in modern cryptography, particularly in algorithms like RSA. It can speed up computations involving large numbers by breaking them down into smaller modular computations, which are then combined using CRT.
A: This calculator is limited by JavaScript's number precision (it uses BigInt for large numbers to overcome this for the main calculation, but intermediate displays might truncate if not carefully handled), and its primary focus on the standard pairwise coprime CRT. Very large numbers or a huge number of congruences might strain browser resources. It does not implement the generalized CRT for non-coprime moduli explicitly (though the underlying math might coincidentally work for some consistent non-coprime cases, it's not guaranteed).
A: In the congruence x ≡ a (mod n):
a (remainder): This is the remainder you get when x is divided by n. It must be a non-negative integer, typically between 0 and n-1.n (modulus): This is the divisor. It must be a positive integer greater than 1. The modulo operation essentially defines a "clock" where numbers "wrap around" after reaching n.Explore more mathematical and computational tools: