Question - Finding Natural Numbers with Four Positive Divisors

To find natural numbers with exactly four positive divisors less than 65, we must identify the form of such numbers. A natural number can have exactly four positive divisors if it is in one of the following forms:

1. The square of a prime number: \( p^2 \)

2. The product of two distinct prime numbers: \( p \times q \)

Now, we will find numbers less than 65 for each case.

1. For \( p^2 \): The prime numbers less than 8 (since \( 8^2 = 64 \)) are 2, 3, 5, and 7.

- \( 2^2 = 4 \)

- \( 3^2 = 9 \)

- \( 5^2 = 25 \)

- \( 7^2 = 49 \)

The valid numbers are 4, 9, 25, and 49.

2. For \( p \times q \): We consider pairs of primes less than 65:

- \( 2 \times 3 = 6 \)

- \( 2 \times 5 = 10 \)

- \( 2 \times 7 = 14 \)

- \( 2 \times 11 = 22 \)

- \( 2 \times 13 = 26 \)

- \( 2 \times 17 = 34 \)

- \( 2 \times 19 = 38 \)

- \( 2 \times 23 = 46 \)

- \( 2 \times 29 = 58 \)

- \( 3 \times 5 = 15 \)

- \( 3 \times 7 = 21 \)

- \( 3 \times 11 = 33 \)

- \( 3 \times 13 = 39 \)

- \( 3 \times 17 = 51 \)

- \( 3 \times 19 = 57 \)

- \( 5 \times 7 = 35 \)

- \( 5 \times 11 = 55 \)

- \( 5 \times 13 = 65 \) (not valid)

- \( 7 \times 11 = 77 \) (not valid)

The valid products are 6, 10, 14, 15, 21, 22, 26, 34, 35, 38, 39, 46, 49, 51, 55, 57, and 58.

The complete list of natural numbers less than 65 with exactly four positive divisors is: 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 34, 35, 38, 39, 49, 51, 55, 57, and 58.

Counting these, we have a total of 18 numbers.

Thus, the answer is 18.