Number of possible value(s) of integer 'a for which the quadratic equation x² + ax +16 =0 has integral roots, is

For integral roots discriminant should be a perfect square.

$\begin{array}{rl} & \therefore {\mathrm{a}}^2-64={\mathrm{\lambda }}^2,\text{where}\mathrm{\lambda }\in \mathrm{Z}\\ & \Rightarrow {\mathrm{a}}^2-{\mathrm{\lambda }}^2=64\\ & \Rightarrow (\mathrm{a}+\mathrm{\lambda })(\mathrm{a}-\mathrm{\lambda })=1\times 64=2\times 32=4\times 16=8\times 8\end{array}$

Since 1 and 64 as factors is not possible. 
The other possible factors are listed below: 

$\begin{array}{rrrrrrr}\mathrm{\alpha }-\mathrm{\lambda }:& 2& -2& 4& -4& 8& -8\\ \mathrm{\alpha }+\mathrm{\lambda }:& 32& -32& 16& -16& 8& -8\\ \mathrm{a}:& 17& -17& 10& -10& 8& -8\end{array}$

Hence, a can take 6 integral values.