K is a 2N$\times$2N symmetric integer matrix with at least one odd element in the diagonal. Suppose $\mathcal{M}$ be a set of integer vectors satisfying the following two properties:
1) $m^{T}K^{-1}m'$ is an integer for any $m,m' \in \mathcal{M}$ .
2) If $l$ is not equivalent to any element of $\mathcal{M}$ then $m^{T}K^{-1}l$ is not an integer for some $m \in \mathcal{M}$.
Let us consider the following set of integer vectors, $\mathcal{L}=\left\{ m+Kx:m\in\mathcal{M},x\in\mathbb{Z}^{2N}\right\}$ . This set forms a 2N-dimensional integer lattice and therefore can be represented as $\mathcal{L}=U\mathbb{Z}^{2N}$, where U is some 2N$\times$2N integer matrix.
Consider the matrix $P\equiv U^{T}K^{-1}U$. Prove that P is an integer matrix and it has at least one odd element on the diagonal.
$$ K = \left( \begin{array}{cc} 3 & 0 \\ 0 & 5 \end{array} \right) $$
$$ U = \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right) $$
$$ U^T K^{-1} U = \left( \begin{array}{cc} \frac{1}{3} & 0 \\ 0 & \frac{1}{5} \end{array} \right) $$