Key idea. The first coordinate must divide 12.

1. From \(xy=12\), we get \(y=12/x\). For both numbers to be positive integers, \(x\) must be a positive divisor of 12.

2. The positive divisors are \(1,2,3,4,6,12\). Substituting each gives

   \[(1,12),(2,6),(3,4),(4,3),(6,2),(12,1).\]

3. Every listed pair has product 12. There can be no other pair, because all possible positive divisors for the first coordinate have been used.

Remember. For ordered pairs, swapping the coordinates usually gives another answer.