Property on the divisor of $n(n+1)/2$

Question

Let $n\geq 0$ and $k\geq n+1$ be a divisor of $\frac{(n+1)(n+2)}{2}$.

Is it true that there exists $k'\geq n$ such that $$0\leq k-k'\leq n+1$$ and $k'$ is a divisor of $\frac{(n+1)n}2$?

I came up with this problem while trying to solve this question. I may be close to a solution but I need the above property to be true, but I failed to prove or disprove it.

Thank you

Answer 1

I believe the statement is false for $n=10$ if $k=33$:

• $k$ is a divisor of $\frac{(n+1)(n+2)}{2}=66$ ✔
• $k\geq n+1$ since $33\geq 11$ ✔

So, what are the possible candidates for $k'$? Well, $k'$ must divide $\frac{n(n+1)}2=55$, so $k$ must be one of $\{1,5,11,55\}$

However, we want $k'<k$, so $55$ is out. But if $k'=11$, then $k-k'=22>11=n+1$, and the same is true for $k'=5$ and $k'=1$.