For any positive integer n, the length of n is defined as the number of prime factors whose product is...

1. Translate the problem requirements: We need to find two-digit numbers (10-99) where the total count of prime factors equals 6. For example, \(75 = 3 \times 5 \times 5\) has length 3 because we count each prime factor occurrence.
2. Identify possible prime factor combinations: Determine all ways to write 6 as a sum of positive integers, where each integer represents how many times a prime appears in the factorization.
3. Calculate minimum values for each combination: For each combination, find the smallest number by using the smallest possible primes, then check if any variations fall within the two-digit range.
4. Count valid two-digit numbers: Systematically check which combinations produce numbers between 10 and 99, counting all possibilities.

Execution of Strategic Approach

1. Translate the problem requirements

When we say the length of a number is the count of prime factors, we're counting each prime factor every time it appears.

For example:

• \(75 = 3 \times 5 \times 5\): length = 3
• \(12 = 2 \times 2 \times 3\): length = 3

We need two-digit numbers (10-99) with exactly 6 prime factors.

2. Identify possible prime factor combinations

List all partitions of 6 and associate with primes:

• \(6 = 6\) → \(2^6\)
• \(6 = 5 + 1\) → \(2^5 \times 3\)
• ... (other partitions) ...

3. Calculate minimum values

• \(2^6 = 64\) ✓
• \(2^5 \times 3 = 96\) ✓
• All others exceed 99 ✗

4. Final Answer

Only two numbers: \(64\) and \(96\).

Answer: 2.