Question 113
Determine all positive integers $M$ for which the sequence $a_0, a_1, a_2, \dots$, defined by $a_0 = \frac{2M+1}{2}$ and $a_{k+1} = a_k \lfloor a_k \rfloor$ for $k = 0, 1, 2, \dots$, contains at least one integer term.
(Remark. For a real number $x$, $\lfloor x \rfloor$ denotes the greatest integer that does not exceed $x$.)
(Remark. For a real number $x$, $\lfloor x \rfloor$ denotes the greatest integer that does not exceed $x$.)