15. Let $a, b \in \mathbf{R}_{+}$, and have the property:
$$
\sqrt{\lg a}+\sqrt{\lg b}+\lg \sqrt{a}+\lg \sqrt{b}=100,
$$

where, $\sqrt{\lg a}, \sqrt{\lg b}, \lg \sqrt{a}, \lg \sqrt{b} \in \mathbf{Z}_{+}$. Then $a b$ $=(\quad)$.
(A) $10^{52}$
(B) $10^{100}$
(C) $10^{144}$
(D) $10^{164}$
(E) $10^{200}$
15. D.

Let $m=\sqrt{\lg a}, n=\sqrt{\lg b}\left(m, n \in \mathbf{Z}_{+}\right)$. Then
$$
\begin{array}{l}
m+n+\frac{1}{2} m^{2}+\frac{1}{2} n^{2}=100 \\
\Rightarrow(m-1)^{2}+(n-1)^{2}=202 \\
\Rightarrow\{m+1, n+1\}=\{11,9\} \\
\Rightarrow\{m+n, m n\}=\{18,80\} .
\end{array}
$$

Therefore, $a b=10^{m^{2}+n^{2}}=10^{(m+n)^{2}-2 m n}=10^{164}$.