Question

A proper divisor of a natural number is a positive integral divisor other than $1$ and the number itself. A natural number greater than $1$ is called “nice” if it equals the product of its distinct proper divisors. What is the sum of the first ten nice numbers?

Answer

Answer: 182

Concepts:factor counting prime factorization

Difficulty rating: 1830

Small Hint:

Express the product of all positive divisors in terms of the number and its divisor count

Big Hint:

The nice numbers are exactly those having four positive divisors

Solution:

If $N$ has $d$ positive divisors, their product is $N^{\frac{d}{2}}.$ Removing $1$ and $N$ leaves product $N^{\frac{d}{2}-1},$ which equals $N$ exactly when $d=4.$ Thus nice numbers are precisely $p^3$ and $pq$ for distinct primes. The first ten are $6,$ $8,$ $10,$ $14,$ $15,$ $21,$ $22,$ $26,$ $27,$ $33,$ whose sum is $182.$

1987 AIME Problem 3

Answer: 182

Difficulty rating: 1830

Small Hint:

Big Hint:

Solution:

Problem 3 in Other Years