6.55 Given $b \in N, a^{100}$ is a 120-digit number, $a^{b}$ is a 10-digit number, then the value of $b$ is
(A) 7.
(B) 8.
(C) 9.
(D) 10.
(Northeast China Three Provinces Mathematics Invitational Competition, 1989)
[Solution] From the given, we know that $119 \leqslant 100 \lg a<120$, which means
$$
1.19 \leqslant \lg a<1.2, \text { and } 9 \leqslant b \lg a<10 \text {, }
$$

then $\frac{9}{1.2}<b<\frac{10}{1.19}$, that is $7.5<b<8.4$.
Since $b \in \mathbb{N}$, then $b=8$.
Therefore, the correct choice is $(B)$.