**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*