1. The smallest positive integer $n=$ that makes $3 n+1$ and $5 n+1$ both perfect squares
Given $3 n+1=k^{2}, 5 n+1=m^{2} \Rightarrow(m+k)(m-k)=2 n$ is even, and since $m+k, m-k$ have the same parity, $n$ must be even. Let $n=2 n_{1} \Rightarrow\left\{\begin{array}{l}m-k=2, \\ m+k=2 n_{1}\end{array} \Rightarrow k=n_{1}-1\right.$. Substituting into $6 n_{1}+1=\left(n_{1}-1\right)^{2} \Rightarrow n_{1}=8$, so the smallest positive integer $n=16$.