Divisor Count Analysis for Number Forms

The question asks to identify which number form among the options does not possess exactly four distinct positive divisors. We will analyze the number of divisors for each form using the divisor function $\tau(n)$.

Number of Divisors Formula

For any positive integer $n$ with prime factorization $n = p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k}$, the total number of distinct positive divisors, denoted as $\tau(n)$, is given by the product of one more than each exponent:

$ \tau(n) = (a_1+1)(a_2+1)\cdots(a_k+1) $

Analyzing Number Forms

Form $p^3$ (for prime $p$)

Consider a number of the form $p^3$, where $p$ is a prime number. The exponents in the prime factorization are just $a_1=3$. Using the formula:

$ \tau(p^3) = (3+1) = 4 $

The distinct positive divisors are $1, p, p^2, p^3$. Thus, this form has exactly four distinct positive divisors.

Form $p \cdot q$ (for distinct primes $p$ and $q$)

Consider a number of the form $p \cdot q$, where $p$ and $q$ are distinct prime numbers. The exponents are $a_1=1$ and $a_2=1$. Using the formula:

$ \tau(p \cdot q) = (1+1)(1+1) = 2 \times 2 = 4 $

The distinct positive divisors are $1, p, q, pq$. Thus, this form also has exactly four distinct positive divisors.

Form $p^2$ (for prime $p$)

Consider a number of the form $p^2$, where $p$ is a prime number. The exponent is $a_1=2$. Using the formula:

$ \tau(p^2) = (2+1) = 3 $

The distinct positive divisors are $1, p, p^2$. Thus, this form has exactly three distinct positive divisors.

Conclusion

Based on the analysis, the number form $p^2$ for a prime $p$ has exactly three distinct positive divisors, while the forms $p^3$ and $p \cdot q$ have exactly four distinct positive divisors. Therefore, $p^2$ is the number that does not have exactly four distinct positive divisors.