Sum
x+y+z=15 ← Equation (1)
Product
xyz=45
z=45xy ← 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=453=15
From Equation (1)
x=0 and y=0 ← not okay
Try xy = 5
z=455=9
From Equation (1)
x=1 and y=5 ← okay
The smallest integer = 1 ← answer