Question

I have a proof question! Any help would be appreciated! Thanks!

Answer 1

Let n∈N have the property that: if a,b∈N and n divides a*b, then either n divides a or n divides b. Show that n is a prime.

Answer 2

whats N?

Answer 3

N refers to the natural numbers.

Answer 4

One way to prove this is via contradiction. Assume that n is not a prime. Then n can be expressed as c*d for some c,d in N, where c,d are both greater than 1. Now, n divides c*d (any number divides itself); however, n cannot divide c or d, since n is greater than c and d. Therefore n cannot be expressed in the form c*d; thus n must be prime. QED.

Answer 5

For example, let n=10, c=2, and d=5. Then 10 divides 2*5, but does not divide 2 or 5.

Answer 6

Thank you so much!

Answer 7

So you don't have to use a or b at all?