Consider the set
$S := \{$ "the number of lies in the set $S$ is a multiple of $n$" $|$ $n = 1, 2, 3, ... , 120 \}$.
Question: Which elements/sentences in the set $S$ are true in every possible non-contradictory case?
Or, in other words:
Does there exist $v$ in $\{0, 1\}^{120}$ satisfying that: the $n$-th component of $v$ is $0$ if and only if the number of $0$ components in $v$ is a multiple of $n$? Which components of $v$ are $0$ if $v$ exists?
$T = \{ n$ divides $|F|: n$ in $\{1,...,120\} \}$ and $F = \{ 1,...,120 \} - T$. Solve the "set equation(s)" for $T$.