[blu42]

"Whenever a product m×n is divisible by p, then m or n must be divisible by p."

Actually it's the other way around:

'Whenever m or n are divisible by p, then their product m×n must be divisible by p.'

The opposite is not true, i.e. the statement is not reversible and still true. For instance:

6x4 = 24; 24 is divisible by 8, i.e. 24 mod 8 = 0, yet 6 mod 8 = 6, and 4 mod 8 = 4.

[ed] I've stepped into a cognition discontinuity, move along, nothing to see.

[jcranmer]

If p is a prime number, then the product m×n being divisible by p means that either m or n is divisible by p. In fact, this is how prime numbers are defined when generalized to generic integral domains.

[blu42]

Doh, my reading balked at the embedded add. Yes, p being a prime actually inverts the statement successfully. My bad.

[camjw]

8 is not prime

[pvg]

"M₆₇ is not a prime" was also a famous single-slide silent lightning talk.

https://en.wikipedia.org/wiki/Frank_Nelson_Cole