  The Hamilton-Waterloo problem asks for a decomposition of the complete graph
into $r$ copies of a 2-factor $F_{1}$ and $s$ copies of a 2-factor $F_{2}$ such
that $r+s=\left\lfloor\frac{v-1}{2}\right\rfloor$. If $F_{1}$ consists of
$m$-cycles and $F_{2}$ consists of $n$ cycles, then we call such a
decomposition a $(m,n)-$HWP$(v;r,s)$. The goal is to find a decomposition for
every possible pair $(r,s)$. In this paper, we show that for odd $x$ and $y$,
there is a $(2^kx,y)-$HWP$(vm;r,s)$ if $\gcd(x,y)\geq 3$, $m\geq 3$, and both
$x$ and $y$ divide $v$, except possibly when $1\in\{r,s\}$.
