Understanding Numbers with Exactly Three Factors

The core idea in this problem involves identifying numbers with precisely three exact factors. A fundamental rule in number theory states that only the squares of prime numbers have exactly three factors. These factors are always 1, the prime number itself, and the square of that prime number.

Let's list the squares of the first few prime numbers to see which ones fit the criteria:

• $2^2 = 4$. The factors are 1, 2, and 4. This number has exactly three factors.
• $3^2 = 9$. The factors are 1, 3, and 9. This number also has exactly three factors.
• $5^2 = 25$. The factors are 1, 5, and 25. This is a number with exactly three factors.
• $7^2 = 49$. The factors are 1, 7, and 49. This number also fits the criteria.
• $11^2 = 121$. The factors are 1, 11, and 121. This is the next number with exactly three factors.

Identifying the Values for p and q

The question specifies that p and q are positive natural numbers possessing exactly three factors. Furthermore, both are two-digit numbers, and it's given that q > p.

From our list above, the two-digit numbers that have exactly three factors are 25 and 49.

Given the condition $q > p$, we assign the values as follows:

• $p = 25$ (the smaller two-digit number with three factors)
• $q = 49$ (the larger two-digit number with three factors)

Identifying the Values for r and s

Similarly, r and s are positive natural numbers with exactly three factors. They are required to be one-digit numbers, with the condition r > s.

Looking again at our list, the one-digit numbers with exactly three factors are 4 and 9.

Applying the condition $r > s$, we assign the values:

• $s = 4$ (the smaller one-digit number with three factors)
• $r = 9$ (the larger one-digit number with three factors)

Calculating the Value of the Expression

We need to compute the value of the expression $\frac{p-q-1}{r-s}$.

Let's substitute the determined values of p, q, r, and s into the expression:

• $p = 25$
• $q = 49$
• $r = 9$
• $s = 4$

The expression becomes:

First, we calculate the numerator:

Next, we calculate the denominator:

Finally, we perform the division:

Therefore, the value of the expression $\frac{p-q-1}{r-s}$ is -5.

Matching the Result with the Given Options

The final step is to compare our calculated value, -5, with the provided options. The options are expressed in terms of $s$, and we found $s=4$. Let's evaluate each option:

• Option 1: $-s - 1$. Substituting $s=4$, we get $-4 - 1 = -5$.
• Option 2: $s - 1$. Substituting $s=4$, we get $4 - 1 = 3$.
• Option 3: $1 - s$. Substituting $s=4$, we get $1 - 4 = -3$.
• Option 4: $s + 1$. Substituting $s=4$, we get $4 + 1 = 5$.

Our calculated result of -5 perfectly matches the value obtained from Option 1.