Answer
The required pair of numbers is $\left( 8,8 \right)$, and their product is $64$.
Work Step by Step
Let us assume the two numbers be x and y. Then,
$x+y=16$ ..... (I)
Now,
$\begin{align}
& x+y=16 \\
& y=16-x \\
\end{align}$
And calculate xy,
$\begin{align}
& xy=x\left( 16-x \right) \\
& xy=16x-{{x}^{2}} \\
\end{align}$
Which is a downwards opening parabola, which attains its maximum value at $\frac{-b}{2a}$ , where $b=16$ and $a=-1$.
$\begin{align}
& x=\frac{-16}{-2} \\
& x=8 \\
\end{align}$
Putting in the value of x in the equation (1) we get,
$\begin{align}
& 8+y=16 \\
& y=16-8 \\
& y=8
\end{align}$
The maximum product is:
$8\times 8=64.$
Hence, the product is maximum when both numbers are 8 and the maximum product is 64.
