If n is not prime, what are the possible values of (n-1)!n ? For which composites is (n-1)!≢0 n

step 1 In this question, we have to answer that if n is a positive integer greater than 1, then is n mi

If n is not prime, what are the possible values of (n-1)!n ?...

Key Concepts

Modular Arithmetic

Modular arithmetic is the study of integers where numbers “wrap around” upon reaching a certain value, the modulus. In this context, when we say that (n–1)! ? r (mod n), it means that when (n–1)! is divided by n, the remainder is r. This concept is foundational for understanding many results in number theory, including congruences and divisibility properties.

Factorial

The factorial of a positive integer k, denoted k!, is the product of all positive integers from 1 up to k. Factorials grow very quickly and often appear in problems involving permutations, combinations, and various number theoretic functions. In this problem, (n–1)! plays a key role in revealing divisibility properties of n.

Composite Numbers

A composite number is an integer greater than 1 that is not prime; that is, it has positive divisors other than 1 and itself. When working with factorials modulo n in the composite case, one often uses the fact that many of the divisors of n will appear in the product (n–1)!, typically forcing (n–1)! to be divisible by n under most circumstances.

Wilson’s Theorem

Wilson’s Theorem states that if p is a prime number, then (p–1)! ? –1 (mod p). The theorem works in one direction to characterize primes and also shows how factorials behave modulo prime numbers. Its converse – that if (n–1)! ? –1 (mod n) then n is prime (with one small exception) – is often used to address which n yield nonzero remainders in factorial congruences.

Divisibility in Factorials

A key idea in this problem is that for most composite numbers n, the factors that make up n appear in the factorial (n–1)! (since n is composite, it has divisors between 1 and n–1). Consequently, (n–1)! usually ends up being divisible by n, leading to (n–1)! ? 0 (mod n). The notable exception is a small composite like 4, where (3)! does not yield 0 modulo 4. This interplay between the factors present in n and those in the factorial is central to understanding the problem.

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