To solve this problem, we need to identify natural numbers between 10 and 500 that have exactly three factors. A number has exactly three factors if and only if it is a square of a prime number.

Explanation:

1. A number $N$ with exactly three factors must be of the form $p^2$, where $p$ is a prime number. This is because:

   • If $N = p^2$, its factors are $1, p,$ and $p^2$, which total three factors.

2. We now find all such $p^2$ between 10 and 500:

   • The smallest prime is $2$: $2^2 = 4$ (too small for our range).
   • The next prime is $3$: $3^2 = 9$ (too small for our range).
   • $5^2 = 25$ (in range).
   • $7^2 = 49$ (in range).
   • $11^2 = 121$ (in range).
   • $13^2 = 169$ (in range).
   • $17^2 = 289$ (in range).
   • $19^2 = 361$ (in range).
   • $23^2 = 529$ (too large for our range).

Conclusion:

The squares of primes between 10 and 500 are $25, 49, 121, 169, 289,$ and $361$. These are 6 numbers in total.

Answer:

There are 6 natural numbers between 10 and 500 that have exactly three factors.

Would you like further clarification or a deeper explanation?

Relative Questions:

1. What is the reasoning behind numbers with three factors being squares of primes?
2. How do we determine whether a number is prime?
3. Can you list other examples of numbers with exactly three factors outside the range of 10 to 500?
4. What are some applications of this concept in number theory?
5. How would the solution change if the range were expanded to 1000?

Tip:

When dealing with problems about factors, always start by understanding the definition and properties of factors, as well as how primes influence factor counts.