Problem 1

Determine all composite integers

n>1

that satisfy the following property: if

d_1, d_2, \ldots, d_k

are all the positive divisors of

n

with

1=d_1<d_2<\cdots<d_k=n

, then

d_i

divides

d_{i+1}+d_{i+2}

for every

1\le i\le k-2

Step 3 of 3: The chain condition forces an impossible ratio

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In plain words

Plugging the mirrored divisors into the required divisibility turns the whole problem into checking whether one specific fraction is an integer, and it never is.

\dfrac{q(p+1)}{p^e}\in\mathbb{Z}

Detailed analysis

Apply the property at index $i=k-e-1$ : $d_{k-e-1}\mid d_{k-e}+d_{k-e+1}$ means $\frac nq \mid \frac n{p^e}+\frac n{p^{e-1}}=\frac{n(p+1)}{p^e}$ . Dividing both sides by $n/p^e$ (the divisibility is equivalent to the resulting ratio being an integer) shows $\frac{q(p+1)}{p^e}$ must be an integer. But $\gcd(p,q)=1$ and $\gcd(p,p+1)=1$ , so $p^e$ cannot divide $q(p+1)$ once $e\ge1$ — a contradiction. Hence $n$ has only one prime factor, so together with Step 1 the composite integers with the property are exactly the prime powers $n=p^r,\ r\ge2$ .