### 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, and then add them up.
To do this, I will try to look for patterns or properties that nice numbers must have.
I notice that if a nice number is prime, then it has no proper divisors, so it cannot be equal to the product of them.
Therefore, nice numbers must be composite.
I also notice that if a nice number has more than two distinct proper divisors, then it cannot be equal to the product of them, because the product would be too large.
For example, if I take 12, which has distinct proper divisors 2, 3, 4, and 6, then the product of them is 2*3*4*6 = 144, which is much larger than 12.
So, nice numbers must have exactly two distinct proper divisors, say a and b.
Then, the nice number must be equal to a*b, and also equal to the product of its proper divisors, which are a and b.
This means that a*b = a*b, which is always true.
Therefore, any composite number with exactly two distinct proper divisors is a nice number.
I can use prime factorization to find such numbers.
If a number has exactly two distinct proper divisors, then it must be the product of two distinct primes, say p and q.
So, nice numbers are of the form p*q, where p and q are distinct primes.
I can start listing the first few nice numbers by multiplying the smallest primes.
The smallest prime is 2, so I can multiply it by the next smallest prime, which is 3.
This gives me the first nice number, 2*3 = 6.
I can keep multiplying 2 by the next primes, such as 5, 7, 11, and so on, to get more nice numbers.
These are 2*5 = 10, 2*7 = 14, 2*11 = 22, and 2*13 = 26.
I already have five nice numbers, and they are all even, because they have 2 as a factor.
I wonder if there are any odd nice numbers.
To get an odd nice number, I would have to multiply two odd primes.
The smallest odd prime is 3, so I can multiply it by the next odd primes, such as 5, 7, 11, and so on, to get more nice numbers.
These are 3*5 = 15, 3*7 = 21, 3*11 = 33, and 3*13 = 39.
I now have nine nice numbers, and I need one more to make ten.
I can try multiplying the next odd prime, which is 17, by either 2 or 3, to get the tenth nice number.
These are 2*17 = 34 and 3*17 = 51.
Either one of them works, but I will choose the smaller one, 34, as the tenth nice number.
Now that I have the first ten nice numbers, I can add them up.
The sum is 6 + 10 + 14 + 22 + 26 + 15 + 21 + 33 + 39 + 34 = 220.