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
