1. Use boiling point elevation formula

   $\Delta T_b = i K_b m$

   Given boiling point of solution $=100.52^\circ C$ and pure water $=100^\circ C$,

   $\Delta T_b = 100.52-100 = 0.52\,K$

2. Calculate molality of the solution

   Mass of solute $\mathrm{AB_2}=10\,g$

   Molar mass of $\mathrm{AB_2}=200\,g\,mol^{-1}$

   $\text{moles of } \mathrm{AB_2} = \frac{10}{200} = 0.05\,mol$

   Mass of water $=100\,g=0.1\,kg$

   $m = \frac{0.05}{0.1} = 0.5\,mol\,kg^{-1}$

3. Find van't Hoff factor $i$

   $0.52 = i \times 0.52 \times 0.5$

   $i = \frac{0.52}{0.26} = 2$

4. Relate $i$ with degree of ionization $\alpha$

   $\mathrm{AB_2 \rightarrow A^{2+} + 2B^-}$

   One formula unit gives total $3$ ions.

   For an electrolyte giving $\nu=3$ ions,

   $i = 1 + (\nu-1)\alpha = 1 + 2\alpha$

   Since $i=2$,

   $2 = 1 + 2\alpha$

   $2\alpha = 1$

   $\alpha = 0.5 = 5 \times 10^{-1}$

5. Nearest integer in the blank

   $\boxed{5}$

The derived answer matches the stored correct answer.