MathDB

Let

n \ge 2

be a positive integer, with divisors

1=d_{1}< d_{2}< \cdots < d_{k}=n \;.

Prove that

d_{1}d_{2}+d_{2}d_{3}+\cdots+d_{k-1}d_{k}

is always less than

n^{2}

, and determine when it divides

n^{2}

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