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The Chinese Remainder Theorem (CRT) is a fundamental theorem in number theory. It allows us to solve systems of simultaneous congruences that have pairwise coprime moduli. For instance, if we have two equations

• \( x \equiv a \, (\text{mod} \, n_1) \)
• \( x \equiv b \, (\text{mod} \, n_2) \)

where \( n_1 \) and \( n_2 \) are coprime, we can find a unique solution modulo \( n_1 \cdot n_2 \). In our example solution, CRT was used to combine the modular congruences after applying Fermat's Little Theorem individually for different moduli (like mod 7 and mod 5). Since each part handled one prime factor, CRT enables deducing the overall congruence under the product of the modulii, ensuring the solution satisfies all given conditions. This theorem simplifies complex modular arithmetic problems into smaller, more manageable ones, making it easier to find solutions that hold true across all given conditions.

The greatest common divisor (gcd) of two numbers is the largest positive integer that divides both numbers without leaving a remainder. When you see \( \operatorname{gcd}(a, b) = 1 \), it means that \( a \) and \( b \) are coprime—they have no common prime factors. This notion is crucial in modular arithmetic as it ensures certain properties, like invertibility. In the exercise, the gcd plays an important role. For example, if \( \operatorname{gcd}(a, 35) = 1 \), then \( a \) is coprime to 35, meaning \( a \) shares no prime factors with 35. Therefore, Fermat's Little Theorem can be applied. It helps demonstrate that certain powers of \( a \) are congruent to 1 mod 35, simplifying complex modular expressions and aiding in conjunction with the Chinese Remainder Theorem to solve congruences.

Modular arithmetic is a mathematical system for integers where numbers "wrap around" upon reaching a certain value, the modulus. It’s like working with a clock: after 12 hours, the time resets back to 1. In essence, when we say something like \( a \equiv b \pmod{n} \), it means \( n \) divides \( a-b \) without a remainder.In the exercise, modular arithmetic is used extensively. For example, when verifying that \( a^{12} \equiv 1 \pmod{35} \), we explore how different powers of \( a \) behave under these constraints, guided by Fermat’s Theorem, which simplifies calculations by reducing powers of \( a \) modulo smaller primes. This system of arithmetic heavily relies on properties like those from Fermat’s Little Theorem, enabling the derivation of reciprocal congruences, and simplifying otherwise daunting problems by decomposing them into more manageable sub-problems.