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Fix $\gamma \in \mathbb{Z}_{>0}^{odd}$ and $n\in\mathbb{Z}_{>0}$. We definethe function $C_\gamma:{\mathbb Z}_{>0}\to {\mathbb Z}_{>0}$ such that if $n$is odd, $C_\gamma(n)=3n+\gamma$; and if $n$ is even, $C_\gamma(n)=n/2$. Wedefine the characteristic mapping $\chi_\gamma: {\mathbb Z}_{>0}\to \{0,\, 1\}$to be $\chi_\gamma(n)\equiv C_\gamma(n)\, {\rm mod}\, 2$. Let $n$ start anintegral loop of length $N$ associated with the $3x+\gamma$ Problem. Let $\rho$and $\nu$ be the count of the number of zeros and ones in a single period of $B= \left(\chi_\gamma^i(n)\right)_{i\geq 0}$. In a single period of $B$, let$m_j$ denote the number of zeros between the $(j-1)$th and $j$th one. Let$\mathcal{M}_n$ be the matrix associated to $n$ whose elements are thesequential products of $2^{m_j}$ (e.g.$(2^{m_0},2^{m_0+m_1},2^{m_0+m_1+m_2},...)$). Let $p$ be a prime factor for allthe terms in the integral loop starting with $n$ with multiplicity $a>0$.Suppose also that $p$ is a prime factor of $\gamma$ and $2^\rho - 3^\nu$ withmultiplicity $b$ and $c$, respectively. Finally assume that $c>b-a$. Then$\det(\mathcal{M}_n) \equiv 0 \pmod p$. We do find examples of this property.Let $\nu$ be prime. Let $z_j = 2^{(m_j+2m_{j+1}+3m_{j+2}+...)/\nu}$ be aweighted arithmetic average of the $m_j$. We prove that if $p$ is a primefactor of $2^\rho-3^\nu$ distinct from $\nu$ so that the residue class $p \pmod\nu$ generates the whole group $\mathbb{Z}_\nu^{*}$ then $p\nmid\det(\mathcal{M}_n)$ if and only if $p\nmid(z_1+...+z_\nu)$ and for any $1\leqi<j\leq \nu$ we have $z_i \not\equiv z_j \pmod p$. By this, we give aninteresting property for the integral loop.

Bonded Trajectories of the $3x+\gamma$ Problem