Let x be one number
Let z be the other number
Their difference is |x-z|
|x-z| = 16

Case 1:               Case 2:
x-z = 16       or     x-z=-16 
 -z = 16-x             -z=-16-x 
  z = -16+x             z=16+x 
  z = x-16              z=x+16
  

Let y be their product zx

y = zx

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In case 1:

y = x(x-16)

y = x²-16x

Vertex formula theorem:

y = ax+bx+c has a minimum value when x = -b/a if a > 0
y = ax+bx+c has a maximum value when x = -b/a if a < 0

y = x²-16x

or

y = 1x²-16x+0

a = 1, b = -16, c = 0

y = 1x-16x+0 has a minimum value when x = -(-16)/1 = 16 since a = 1 > 0

z = 16-16 = 0

So the numbers are 16 and 0

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In case 2:

y = x(x+16)

y = x²+16x

Vertex formula theorem:

y = ax+bx+c has a minimum value when x = -b/a if a > 0
y = ax+bx+c has a maximum value when x = -b/a if a < 0

y = x²+16x

or

y = 1x²+16x+0

a = 1, b = 16, c = 0

y = 1x+16x+0 has a minimum value when x = -16/1 = -16 since a = 1 > 0

z = x+16 = -16+16 = 0

So the numbers are 0 and -16 

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So there are two possibilities,

The two numbers could be 16 and 0 or else they could be 0 and -16

Edwin