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The correct answer is $2$. An equation with one variable, $x$, has infinitely many solutions only when both sides of the equation are equal for any defined value of $x$. It's given that $2x+16=a\left(x+8\right)$, where $a$ is a constant. This equation can be rewritten as $2\left(x+8\right)=a\left(x+8\right)$. If this equation has infinitely many solutions, then both sides of this equation are equal for any defined value of $x$. Both sides of this equation are equal for any defined value of $x$ when $2=a$. Therefore, if the equation has infinitely many solutions, the value of $a$ is $2$.

Alternate approach: If the given equation, $2x+16=a\left(x+8\right)$, has infinitely many solutions, then both sides of this equation are equal for any value of $x$. If $x=0$, then substituting $0$ for $x$ in $2x+16=a\left(x+8\right)$ yields $2\left(0\right)+16=a\left(0+8\right)$, or $16=8a$. Dividing both sides of this equation by $8$ yields $2=a$.