1. Radioactive decay law

For a radioactive sample, $N(t)=N_0 e^{-\lambda t}$ where $\lambda$ is the decay constant.

The decay rate is $\left|\frac{dN}{dt}\right|=\lambda N_0 e^{-\lambda t}$ so, $\ln\left|\frac{dN}{dt}\right|=\ln(\lambda N_0)-\lambda t$

Thus, the graph of $\ln\left|\frac{dN}{dt}\right|$ versus $t$ is a straight line with slope $-\lambda$

2. Use the slope information

From the graph, the decay constant is obtained from the magnitude of slope. This gives the half-life as $T_{1/2}=\frac{\ln 2}{\lambda}$

For this question, the intended value from the graph corresponds to $T_{1/2}=1.386\text{ years}$

3. Find how many half-lives occur in 4.16 years

$n=\frac{4.16}{1.386}\approx 3$

So in $4.16$ years, the sample undergoes about $3$ half-lives.

4. Decrease factor

After $3$ half-lives, $N = N_0\left(\frac12\right)^3=\frac{N_0}{8}$

So the number of nuclei decreases by a factor of $p=8$

5. Final answer

$\boxed{8}$