### Instruction: By a proper divisor of a natural number we mean a positive integral divisor other than 1 and the number itself. A natural number greater than 1 will be called nice if it is equal to the product of its distinct proper divisors. What is the sum of the first ten nice numbers?### Response: I need to find the first ten natural numbers that are equal to the product of their distinct proper divisors.
For example, 6 is a nice number, because its proper divisors are 2 and 3, and 6 = 2 * 3.
But 8 is not a nice number, because its proper divisors are 2 and 4, and 8 is not equal to 2 * 4.
I notice that a nice number must have at least two distinct proper divisors, so it cannot be prime or a power of a prime.
I also notice that if a number has more than two distinct proper divisors, then the product of those divisors will be larger than the number itself.
For example, 12 has proper divisors 2, 3, 4, and 6, and 2 * 3 * 4 * 6 = 144 > 12.
So a nice number can only have exactly two distinct proper divisors.
That means a nice number must be the product of two distinct primes, like 6 = 2 * 3.
I can list the first few pairs of distinct primes: (2, 3), (2, 5), (2, 7), (3, 5), (3, 7), (2, 11), (3, 11), (5, 7), (2, 13), (3, 13).
The product of each pair is a nice number: 6, 10, 14, 15, 21, 22, 33, 35, 26, 39.
These are the first ten nice numbers, in ascending order.
To find their sum, I can add them up: 6 + 10 + 14 + 15 + 21 + 22 + 33 + 35 + 26 + 39 = 221.