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ScanSolve AI Answer
Nike🙃, let's knock this out. The Chinese Remainder Theorem (CRT) provides a way to solve systems of simultaneous congruences with different moduli. It states that if you have a set of congruences: x ≡ a₁ (mod m₁) x ≡ a₂ (mod m₂) ... x ≡ aₖ (mod mₖ) where m₁, m₂, ..., mₖ are pairwise coprime (i.e., the greatest common divisor of any two is 1), then there exists a unique solution x modulo M, where M = m₁ × m₂ × ... × mₖ. To solve a problem using the CRT, follow these steps: 1. Calculate M: Find the product of all moduli, M = m₁ × m₂ × ... × mₖ. 2. Find Mᵢ: For each i, calculate Mᵢ = M / mᵢ. 3. Find the modular inverse: For each i, find the inverse of Mᵢ modulo mᵢ, denoted as yᵢ, such that Mᵢ × yᵢ ≡ 1 (mod mᵢ). 4. Construct the solution: The solution is given by: x ≡ Σ (aᵢ × Mᵢ × yᵢ) (mod M) 5. Simplify: Reduce x modulo M to find the smallest non-negative solution. This theorem is particularly useful in number theory and cryptography. If you have a specific problem to apply this to, please share it! Send me the next one 📸