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Given two finite sets of integers $S\subseteq\NNN\setminus\{0\}$ and$D\subseteq\NNN\setminus\{0,1\}$,the impartial combinatorial game $\IMARK(S,D)$is played on a heap of tokens. From a heap of $n$ tokens, each player canmoveeither to a heap of $n-s$ tokens for some $s\in S$, or to a heap of $n/d$tokensfor some $d\in D$ if $d$ divides $n$.Such games can be considered as anintegral variant of \MARK-type games, introduced by Elwyn Berlekamp and JoeBuhlerand studied by Aviezri Fraenkel and Alan Guo, for which it is allowed tomove from a heap of $n$ tokensto a heap of $\lfloor n/d\rfloor$ tokens for any$d\in D$.Under normal convention, it is observed that the Sprague-Grundysequence of the game $\IMARK(S,D)$ is aperiodic for any sets $S$ and$D$.However, we prove that, in many cases, this sequence is almost periodic andthat the set of winning positions is periodic.Moreover, in all these cases, theSprague-Grundy value of a heap of $n$ tokens can be computed in time $O(\logn)$.We also prove that, under mis\`ere convention, the outcome sequence ofthese games is purely periodic.

i-MARK: A New Subtraction Division Game