= Solution
For a <hyperfine multiplet> governed by the <magnetic-dipole hyperfine interaction>, adjacent <frequency> gaps obey $\nu_F=|A|F/h$. The five measured gaps for the first isotope are approximately in the ratios $1:2:3:4:5$. Therefore the six levels have $F=0,1,2,3,4,5$: the first measured gap corresponds to higher $F=1$. In particular $F_{\min}=|I-J|=0$ and $F_{\max}=I+J=5$, giving
$$
\boxed{I=J=5/2.}
$$
This inference uses both the number of intervals and their successive ratios. Merely seeing six levels would specify only $\min(I,J)=5/2$ and would not by itself establish equality.
Solving a <linear least-squares problem> for a fit through the origin, taking the rounded <frequencies> as equally weighted, gives
$$
\boxed{\frac{|A_{55}|}{h}=\frac{\sum_{F=1}^5F\nu_F}{\sum_{F=1}^5F^2}
=72.42\ \mathrm{MHz}\quad\text{approximately}.}
$$
Thus the scale of the <hyperfine structure> is about $4.80\times10^{-26}$ J, or $3.00\times10^{-7}$ eV. The displayed <frequencies> have unequal rounding precision, so additional fit digits are not physically warranted. The gap pattern supports a leading dipole interval rule, with small residuals that do not establish higher-multipole constants without uncertainties.
The total width between $F=0$ and $F=5$ is $15|A_{55}|$, about $1.086$ GHz in <frequency> units. The sign of $A_{55}$, the absolute <nuclear g-factor> and the electronic field coefficient cannot be separated from these gap magnitudes alone. Measuring the level ordering or using another known magnetic property would supply additional information. The figure in <2/b> shows the positive-$A$ ordering as an illustration.
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