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Prime numbers are the building blocks of arithmetic. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. This means:

• The number must be greater than 1.
• It can only be divided evenly by 1 and itself.

Some examples of prime numbers are 2, 3, 5, 7, 11, and 13. Importantly, 2 is the only even prime number. Every number after 2 that is prime must be odd, because any even number can be divided by 2.

Determining whether a number is prime can become difficult as numbers get larger, which is where the use of Wilson's Theorem can come in handy. This theorem is particularly useful because it provides a clear method to confirm the primality of numbers. According to this theorem, for a number \( p \) to be prime, \( (p-1)! + 1\) must be divisible by \( p \). But essentially it means that \( (p-1)! \equiv -1 \pmod{p} \).

Modular arithmetic is like clockwork arithmetic. It deals with integers and the remainder when one number is divided by another, known as the modulus. Imagine you're on a clock: after reaching 12, it starts again at 1. Similarly, if we have a modulus of 5, after we reach 5, we cycle back to 0.

Using modular arithmetic, calculations are simplified into a specific range—0 to one less than the modulus. For example:

• \(7 \mod 5 = 2\) because dividing 7 by 5 gives a remainder of 2.
• \(18 \mod 5 = 3\) since 18 divided by 5 leaves a remainder of 3.

This concept is highly useful in simplifying large number calculations. In the exercise, instead of directly calculating \(16!\), modular arithmetic allows us to simplify and check conditions defined by Wilson's Theorem without excessive computation.

The factorial of a natural number \(n\), denoted as \(n!\), is the product of all positive integers less than or equal to \(n\). For instance, \(4! = 4 \times 3 \times 2 \times 1 = 24\).

The concept of factorial is vital in permutations and combinations to determine the number of ways to arrange\, or combine things. Factorials grow very quickly, for instance:

• \(5! = 120\)
• \(6! = 720\)
• \(10! = 3,628,800\)

In the context of Wilson’s Theorem, we use factorials to test for primality. The theorem checks if the factorial of one less than a given number \((p-1)!\) is congruent to \(-1\) modulo \(p\). If so, then \(p\) is prime. In our example, \(16!\) needs to be checked mod \(17\) for a result of \(-1\) to confirm that 17 is indeed prime.