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A pseudoprime is a composite number that can mimic the behavior of a prime according to Fermat's Little Theorem for some base. Basically, if for a base \(a\), a number \(n\) satisfies \(a^{n-1} \equiv 1 \pmod{n}\), we might think \(n\) is prime. However, if \(n\) doesn't fulfill primality but the equation holds true for some base, it is called pseudoprime in that base.
Pseudoprimes can be misleading during prime testing, so we use them carefully in number theory. For example, 341 is a pseudoprime to base 2 because \(2^{340} \equiv 1 \pmod{341}\) while 341 is not a prime number because it equals 11 \(\times\) 31.
Notably, showing a number is not a pseudoprime involves finding a base for which the Fermat test fails, as seen in the example with 12.

Congruence is a relation stating two numbers have the same remainder when divided by a given number (called modulus). For instance, considering 32 and 1 modulo 12, we say 32 \(\equiv\) 1 (mod 12) because dividing both numbers by 12 results in the same remainder, which is 8.
This concept is foundational in various number theory theorems and proofs like Fermat's Little Theorem. It helps us deeply understand how numbers relate to each other under modular arithmetic.
Recognizing congruences allows for easier calculations in many mathematical problems, maintaining coherence throughout solutions.

Two numbers are coprime if their greatest common divisor (GCD) is 1. This means they do not share any common factors other than 1.
Coprime numbers have unique properties and are helpful in tests like Fermat's, ensuring the base does not divide the number in question. For instance, 2 and 12 are coprime because their GCD is 1, making 2 a valid base to test whether 12 is a pseudoprime.
Identifying coprime pairs is essential for understanding prime characteristics and solving congruences. It also aids in simplifying fractions and solving Diophantine equations effectively.