We solve this system using the Chinese Remainder Theorem. Since 36 and 7 are coprime, there is a unique solution modulo $36 \cdot 7 = 252$.

["Solving Integer Systems with the Chinese Remainder Theorem: A Unique Solution Exists When 36 and 7 Are Coprime", "When faced with simultaneous congruences in number theory, the Chinese Remainder Theorem (CRT) offers a powerful and elegant solution. One classic application is solving systems of congruences where the moduli are coprime—ensuring a unique solution modulo the product of the moduli.", "Consider the system:", "$$\n\begin{cases}\nx \equiv a \pmod{36} \\nx \equiv b \pmod{7}\n\end{cases}\n$$", "where 36 and 7 are coprime integers—specifically, they share no common prime factors. According to the Chinese Remainder Theorem, since $\gcd(36, 7) = 1$, there exists a unique integer $x$ satisfying both congruences, and the solution is guaranteed modulo $36 \ imes 7 = 252$.", "### Why the Theorem Works", "The theorem makes sense mathematically because the congruences define consistent residue conditions across two independent moduli. Since moduli 36 and 7 are relatively prime, the combined system captures a complete residue solution across the full range from 0 to 251 (inclusive).", "This method efficiently reconstructs an integer $x$ such that:", "- $x \mod 36 = a$\n- $x \mod 7 = b$", "No trial and error is required—CRT delivers a precise, closed-form solution within a well-defined range.", "### How the Solution Is Constructed", "To compute the solution formally, we express $x$ in the form:", "$$\nx = a + 36k\n$$", "for some integer $k$. Substituting into the second congruence gives:", "$$\na + 36k \equiv b \pmod{7}\n\Rightarrow 36k \equiv b - a \pmod{7}\n$$", "Since $36 \mod 7 = 1$ (because $36 = 5 \cdot 7 + 1$), this simplifies to:", "$$\nk \equiv b - a \pmod{7}\n\Rightarrow k = b - a + 7m \quad \ ext{for some integer } m\n$$", "Plugging back gives:", "$$\nx = a + 36(b - a + 7m) = a + 36(b - a) + 252m\n\Rightarrow x \equiv a + 36(b - a) \pmod{252}\n$$", "Thus, the unique solution modulo 252 is:", "$$\nx \equiv 36(b - a) + a \pmod{252}\n$$", "This formula delivers the exact value of $x$ satisfying both congruences in a systematic, efficient way.", "### Real-World Applications", "The Chinese Remainder Theorem is not just a theoretical curiosity—it’s widely used in cryptography (e.g., RSA decryption), scheduling problems, error-correcting codes, and distributed computing. Its ability to fuse modular information into a single coherent residue class makes it indispensable wherever consistent multi-modulus information must be reconciled.", "### Conclusion", "In summary, when working with simultaneous congruences whose moduli are coprime—like 36 and 7—we leverage the Chinese Remainder Theorem to guarantee a unique solution modulo their product, 252. This avoids ambiguity, streamlines computation, and enhances reliability in mathematical modeling and algorithmic design.", "This elegant framework underscores why CRT remains a cornerstone of number theory and applied mathematics.", "---", "Unlock the power of modular arithmetic: solve systems efficiently with the Chinese Remainder Theorem. Explore new dimensions in computational number theory today."]