  Let $k\ge 0,a\ge 1$ and $b\ge 0$ be integers. We define the arithmetic
function $g_{k,a,b}$ for any positive integer $n$ by
$g_{k,a,b}(n):=\frac{(b+na)(b+(n+1)a)...(b+(n+k)a)} {{\rm
lcm}(b+na,b+(n+1)a,...,b+(n+k)a)}.$ Letting $a=1$ and $b=0$, then $g_{k,a,b}$
becomes the arithmetic function introduced previously by Farhi. Farhi proved
that $g_{k,1,0}$ is periodic and that $k!$ is a period. Hong and Yang improved
Farhi's period $k!$ to ${\rm lcm}(1,2,...,k)$ and conjectured that $\frac{{\rm
lcm}(1,2,...,k,k+1)}{k+1}$ divides the smallest period of $g_{k,1,0}$.
Recently, Farhi and Kane proved this conjecture and determined the smallest
period of $g_{k,1,0}$. For the general integers $a\ge 1$ and $b\ge 0$, it is
natural to ask the interesting question: Is $g_{k,a,b}$ periodic? If so, then
what is the smallest period of $g_{k,a,b}$? We first show that the arithmetic
function $g_{k,a,b}$ is periodic. Subsequently, we provide detailed $p$-adic
analysis of the periodic function $g_{k,a,b}$. Finally, we determine the
smallest period of $g_{k,a,b}$. Our result extends the Farhi-Kane theorem from
the set of positive integers to general arithmetic progressions.
