9. 1. $2^{3} \times 3 \times 7=168$.

Let the two simplest fractions be $\frac{a}{600}$ and $\frac{b}{700}$. Then $(a, 6)=(b, 7)=1$.
Thus, the sum $\frac{7 a+6 b}{4200}$ has a numerator that is coprime with 6 and 7.
Since $4200=2^{3} \times 3 \times 7 \times 5^{2}$, after canceling out the common factors, the denominator is no less than $2^{3} \times 3 \times 7=168$.
On the other hand, such a denominator can be obtained:
$$\frac{1}{600}+\frac{3}{700}=\frac{1}{168} \text {. }$$