Chinese Remainder Theorem Calculator Negatives

The Chinese Remainder Theorem (CRT) provides a way to solve systems of congruences with pairwise coprime moduli. This calculator extends the standard CRT to handle negative numbers in congruences.

What is the Chinese Remainder Theorem?

The Chinese Remainder Theorem is a fundamental result in number theory that states that if the moduli are pairwise coprime, then for any given set of remainders, there exists a unique solution modulo the product of the moduli.

In its basic form, the theorem solves systems of congruences like:

x ≡ a₁ mod m₁ x ≡ a₂ mod m₂ ... x ≡ aₙ mod mₙ

where m₁, m₂, ..., mₙ are pairwise coprime integers. The solution x is unique modulo the product of all moduli (m₁ × m₂ × ... × mₙ).

This calculator extends the theorem to handle negative remainders and moduli, which is useful in cryptography and other mathematical applications.

How to Use This Calculator

To use the calculator:

Enter the number of congruences you want to solve (minimum 2, maximum 5)
For each congruence, enter:
- The remainder (can be negative)
- The modulus (must be positive)
Click "Calculate" to find the solution
Review the result and chart visualization

Note: All moduli must be pairwise coprime for the theorem to guarantee a unique solution. The calculator will check this condition.

The Formula

The Chinese Remainder Theorem provides a formula to find the solution x to the system of congruences:

x ≡ a₁ mod m₁ x ≡ a₂ mod m₂ ... x ≡ aₙ mod mₙ

The solution can be found using the following steps:

Compute M = m₁ × m₂ × ... × mₙ
For each i from 1 to n:
- Compute Mᵢ = M / mᵢ
- Find the modular inverse of Mᵢ modulo mᵢ (denoted as yᵢ)
Compute x = (a₁ × M₁ × y₁ + a₂ × M₂ × y₂ + ... + aₙ × Mₙ × yₙ) mod M

This calculator implements this algorithm to solve systems with negative remainders.

Worked Example

Let's solve the following system of congruences:

x ≡ 2 mod 3 x ≡ -1 mod 5 x ≡ 4 mod 7

Step 1: Verify that the moduli are pairwise coprime (3, 5, and 7 are all prime and distinct)

Step 2: Compute M = 3 × 5 × 7 = 105

Step 3: For each congruence:

For x ≡ 2 mod 3:
- M₁ = 105 / 3 = 35
- y₁ is the inverse of 35 mod 3, which is 2 (since 35 ≡ 2 mod 3)
For x ≡ -1 mod 5:
- M₂ = 105 / 5 = 21
- y₂ is the inverse of 21 mod 5, which is 1 (since 21 ≡ 1 mod 5)
For x ≡ 4 mod 7:
- M₃ = 105 / 7 = 15
- y₃ is the inverse of 15 mod 7, which is 1 (since 15 ≡ 1 mod 7)

Step 4: Compute x:

x = (2 × 35 × 2 + (-1) × 21 × 1 + 4 × 15 × 1) mod 105 x = (140 - 21 + 60) mod 105 x = 179 mod 105 x = 74

The solution is x = 74, which satisfies all three original congruences.

Frequently Asked Questions

Can I use negative numbers in the congruences?

Yes, this calculator accepts negative numbers for both remainders and moduli. The Chinese Remainder Theorem works with negative numbers as long as the moduli are positive and pairwise coprime.

What happens if the moduli aren't pairwise coprime?

The Chinese Remainder Theorem requires that all moduli are pairwise coprime. If they aren't, the system may have no solution or infinitely many solutions. The calculator will check this condition and display an appropriate message.

How does the calculator handle negative solutions?

The calculator will return the smallest positive solution by default. You can adjust the range if needed, but the theorem guarantees a unique solution modulo the product of the moduli.

Can I use this calculator for cryptography?

Yes, the Chinese Remainder Theorem is used in various cryptographic algorithms. This calculator can help you understand and implement CRT-based systems with negative numbers.