Question

Given a composite number m > 4, it implies that:

a) (m-1)! ≡ 1 (mod m)
b) (m-1)! ≡ 0 (mod m)
c) (m-1)! ≡ m (mod m)
d) (m-1)! ≡ -1 (mod m)

Answer 1

For a composite number m > 4, the factorial (m-1)! will contain m as a factor, making (m-1)! congruent to 0 modulo m.

Step-by-step explanation:

The question asks about the congruence of the factorial of a composite number m minus one, denoted as (m-1)!, modulo m. For a composite number m > 4, one or more factors of (m-1)! will be the factors of m itself. This means that the product (m-1)! will contain m as a factor, hence (m-1)! ≃ 0 (mod m). So the correct answer is (b) (m-1)! ≃ 0 (mod m).