We are tasked with finding the number of ways to choose three positive integers $a, b, c$ such that:

1. $a < b < c$,
2. $a \cdot b \cdot c = 100$.

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Step 1: Prime factorization of $100$

The prime factorization of $100$ is: $100 = 2^2 \cdot 5^2.$

Thus, any $a, b, c$ must be factors of $100$, and their product must still equal $100$.

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Step 2: List the factors of $100$

The factors of $100$ are: $1, 2, 4, 5, 10, 20, 25, 50, 100.$

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Step 3: Find distinct triplets $(a, b, c)$

We now determine all possible sets of three factors $(a, b, c)$ where $a < b < c$, and their product equals $100$.

Case-by-case enumeration:

1. Case 1: $a = 1$ $b \cdot c = 100 \quad \text{with } b < c.$

   • $b = 2, c = 50 \quad (1 \cdot 2 \cdot 50 = 100)$,
   • $b = 4, c = 25 \quad (1 \cdot 4 \cdot 25 = 100)$,
   • $b = 5, c = 20 \quad (1 \cdot 5 \cdot 20 = 100)$,
   • $b = 10, c = 10 \quad \text{(invalid since } b \neq c)$.

   Valid triplets: $(1, 2, 50), (1, 4, 25), (1, 5, 20)$.

2. Case 2: $a = 2$ $b \cdot c = 50 \quad \text{with } b < c.$

   • $b = 5, c = 10 \quad (2 \cdot 5 \cdot 10 = 100)$.

   Valid triplet: $(2, 5, 10)$.

3. Case 3: $a = 4$ $b \cdot c = 25 \quad \text{with } b < c.$

   • $b = 5, c = 5 \quad \text{(invalid since } b \neq c)$.

   No valid triplets.

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Step 4: Count the number of triplets

The valid triplets are: $(1, 2, 50), (1, 4, 25), (1, 5, 20), (2, 5, 10).$

Thus, there are 4 ways to choose $(a, b, c)$ such that $a < b < c$ and $a \cdot b \cdot c = 100$.

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Final Answer:

4

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Would you like me to clarify any step or expand further?

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Follow-up Questions:

1. How do we handle cases where the integers are not distinct?
2. What happens if one of the integers is fixed, e.g., $a = 2$?
3. How does the solution change for a different product, such as $a \cdot b \cdot c = 200$?
4. Can you explain the reasoning behind using prime factorization in problems like this?
5. How can this approach be extended to solve for four or more integers?

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Tip:

When solving factorization problems with constraints, always start by listing all factors and work systematically to avoid missing cases.