Question 914653
Suppose the proper divisors of n are *[tex \large d_1, \ldots, d_k] where *[tex \large 1 < d_1 < \ldots < d_k < n]. Then we have *[tex \large d_1d_k = n], *[tex \large d_2d_{k-1} = n], and so on. We want the product of the d_i's to equal n, and this occurs if and only if n has two proper divisors (excluding 1), or when n has exactly four divisors. This occurs if and only if n is a product of two different primes, or n is the cube of a prime.


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