Sum
$x + y + z = 15$   ←   Equation (1)
 

Product
$xyz = 45$

$z = \dfrac{45}{xy}$   ←   Equation (2)
 

For z to be a positive integer, the possible values of xy are 1, 3, 5, 9, and 45.
 

Try xy = 3

$z = \dfrac{45}{3} = 15$
 

From Equation (1)
$x = 0 \text{ and } y = 0$   ←   not okay

 

Try xy = 5

$z = \dfrac{45}{5} = 9$
 

From Equation (1)
$x = 1 \text{ and } y = 5$   ←   okay

 

The smallest integer = 1   ←   answer