Exercise
Suppose a is a nonzero integer. If for all integers b and c we have that a | bc implies a divides b or c, then which of the following is true?
a is a prime
a is a prime or 1
|a| is a prime or 1
No, a = 1 is also a possibility
No, a = -1 is also a possibility
Indeed, suppose |a| > 1, and d is divisor of
a. Then |a| = |d| · |a/d|.
Thus a divides |d| or |a/d|. From this we deduce that
|d| = 1 or |a|, which then implies that |a| is prime