9) Anyway, if you take a natural number $n$, the number $(n+2)(n+3)(2n+5)$ is divisible by:
(A) 4, (B) 6,
(C) 9
(D) 10
(E) 15.
9. The answer is (B). Let $n$ be any natural number and let $N=(n+2)(n+3)(2 n+5) . \quad N$ is divisible by two consecutive numbers: $n+2$ and $n+3$, so it is certainly divisible by an even number and therefore is divisible by 2. Now let's prove that $N$ is also divisible by 3 and therefore is divisible by 6. If $n$ is divisible by 3, then so is $n+3$ and therefore $N$ is. If $n$ divided by 3 gives a remainder of 1, then $n+2$ is divisible by 3 and therefore $N$ is divisible by 3. Finally, if $n$ divided by 3 gives a remainder of 2, then $n$ can be written in the form $n=3 k+2$ for some natural number $k$. So $2 n+5=6 k+4+5=3(2 k+3)$ is divisible by 3 and therefore so is $N$. We also observe that the answers (A), (C), (D), and (E) are not correct: indeed, if $n=1, N=84$ which is not divisible by 9, 10, nor 15; if $n=3, N=330$ which is not divisible by 4.