7. Given $a, b, c, d \in \mathbf{N}$ and satisfy $342(a b c d+$ $a b+a d+c d+1)=379(b c d+b+d)$, let $M=a$ $\cdot 10^{3}+b \cdot 10^{2}+c \cdot 10+d$ then the value of $M$ is $\qquad$ .
7. 1949.

Solution: $\because$ From the given, we have $1+\frac{1}{9+\frac{1}{4+\frac{1}{9}}}=a+$ $\frac{1}{b+\frac{1}{c+\frac{1}{d}}}$, the continued fraction representation is unique.
8. 11
Solution: $\because\left(a_{1}+a_{2}+\cdots+a_{99}\right)^{2}=a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2}$ $+2 \sum_{1 \in i<j \leqslant 99} a_{i} a_{j} \Rightarrow\left(a_{1}+\cdots a_{n}\right)^{2}=99+2 \sum$ to make $99+$ $2 \sum$ a perfect square, the smallest positive integer is 121, i.e., when taking 55 ones and 44 negative ones, or 55 negative ones and 44 ones, we can have $a_{1}+\cdots+$ $a_{99}=11 \quad \therefore \sum$'s minimum value is 11.