Let $A$ be an 8 by 8 integral unimodular matrix. Moreover, assume that for each $x \in \mathbb{Z}^8$, we have $x^{\top} A x$ is even. What is the possible number of positive eigenvalues for $A$?

Let $\varphi(x)$ be the Euler’s totient function. Let $S = \{a_1,\dots, a_n\}$ be a set of positive integers such that for any $a_i$, all of its positive divisors are also in $S$. Let $A$ be the matrix with entries $A_{i,j} = gcd(a_i,a_j)$ being the greatest common divisors of $a_i$ and $a_j$. Prove that $\det(A) = \prod_{i=1}^{n} \varphi(a_i)$.

Let $S$ be the set of all 4 by 4 integral positive-definite symmetric unimodular matrices. Define an equivalence relation $\sim$ on $S$ such that for any $A,B \in S$, we have $A \sim B$ if and only if $PAP^\top = B$ for some integral unimodular matrix $P$. Determine $S ~/\sim$.