= Solution
Identifying the fitted limiting extension with the <polymer contour length> gives
$$
\boxed{N\simeq\frac{500\,\mathrm{nm}}{0.64\,\mathrm{nm}}=781.25\simeq7.8\times10^2}.
$$
These are statistical links of the <freely jointed chain>, not necessarily chemical <polymer repeat units>. A measured maximum just before detachment need not equal the true contour length exactly, so the identification is itself an estimate.
The stated temperature request mixes two distinct quantities. At unchanged $N$ and $b$, the true <freely jointed chain> maximum is geometric:
$$
\boxed{r_{\max}(100^\circ\mathrm C)=Nb\simeq500\,\mathrm{nm}}.
$$
In contrast, the low-force relation $\langle z\rangle=Nb^2f/(3k_BT)$ predicts a ratio at the \i[same small force], not a new maximum:
$$
\boxed{\frac{\langle z\rangle_{373.15}}{\langle z\rangle_{298.15}}
=\frac{298.15}{373.15}\simeq0.799}.
$$
Formally multiplying $500\,\mathrm{nm}$ by this ratio gives about $400\,\mathrm{nm}$, but $500\,\mathrm{nm}=Nb$ is outside the small-extension domain, so that is not a justified maximum-extension prediction. A concrete internally consistent counterexample uses the same chain at $f=0.10\,\mathrm{pN}$: the low-force extensions are about $2.59\,\mathrm{nm}$ and $2.07\,\mathrm{nm}$, while both contour lengths remain $500\,\mathrm{nm}$. Any temperature dependence of the segment length, solvent interactions, attachment strength or rupture force needs information absent from the model.
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