Problem 4

Let

n\ge2

be a positive integer with divisors

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

. Prove that

d_1d_2+d_2d_3+\cdots+d_{k-1}d_k

is always less than

n^2

, and determine when it is a divisor of

n^2

Step 3 of 3: Classify divisibility

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In plain words

Classify divisibility

\frac{n^2}{p}<S<n^2

Detailed analysis

For prime $n$ , $S=n$ works. If composite and $p$ is its least prime divisor, then $d_{k-1}=n/p$ and $S>n^2/p$ , while $n^2/p$ is the largest proper divisor of $n^2$ ; thus $S$ cannot divide $n^2$ .