I have a statement There is a natural number, whose every divisor is smaller than 147. and cannot figure out, which of the following options is correct (it can be more of them)
a.) $(∃x∈N)(∃n∈N)(∃k∈N)((x=n⋅k) ∧ (n<147))$
b.) $(∃x∈N)(∀n∈N)(∀k∈N)((x=n⋅k) ∨ (n<147))$
c.) $(∃x∈N)(∀n∈N)(∀k∈N)(x=n⋅k ⇒ n<147)$
d.) $(∃x∈N)(∃n∈N)(∀k∈N)(x=n⋅k ⇒n <147)$
a) says that there is some number with some divisor smaller than 147
b) is really weird ... It says that there is some number $x$ such that for every number $n$: $n$ is either smaller than $147$, or it is true that for all numbers $k$: $x=n\cdot k$. So note that if you pick some number $n \geq$ (say, $n=1785$) it must then be true that for all numbers $k$: $x=1785\cdot k$. That's a crazy claim: it would mean that $x=1785 \cdot 1 = 1785 \cdot 2 = 1783 \cdot 3 ...$ which is clearly false. But the claim you have to symbolize is clearly true: Sure, there is some number such that all of its divisors are smaller than $147$: this is clearly true for any number smaller than $147$. So, no, b) is not it either
c) is right: If $n$ is a divisor of $x$, then $n$ is smaller than $147$
d) is bad news: You have an existential in combination with a conditional, which is almost always a red flag, because you can always make a conditional trivially true by making its antecedent false. So it is here: Can you find some number $n$ such that if $x = n \cdot k$, then $n <147$? Yes, exactly because you can simply pick $n > x$, in which case $x = n \cdot k$ is false, and hence $x = n \cdot k \rightarrow n < 147$ is true. But this is what is called a 'vacuous truth'... it does not make any interesting claim at all. It certainly does not end up saying that every divisor of $x$ has to be smaller than $147$