The question involves using the digits 8, 4, 7, 3, 0 to fill in the boxes in two separate multiplication problems. The goal is:

1. $\alpha$: To maximize the product.
2. $\beta$: To minimize the product.

The process requires strategic placement of the digits in the two numbers being multiplied.

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Solution for $\alpha$: Maximize the product

To maximize the product, you should distribute the digits to form the largest possible numbers. Larger numbers have a more significant effect on the product.

Approach:

• Form the largest possible two-digit and three-digit numbers from the digits: 873 × 40 (or 870 × 43).

Maximized Product:

$873 × 40 = 34,920$

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Solution for $\beta$: Minimize the product

To minimize the product, you should distribute the digits to form the smallest possible numbers. Smaller numbers yield a smaller product.

Approach:

• Form the smallest possible two-digit and three-digit numbers from the digits: 30 × 478.

Minimized Product:

$30 × 478 = 14,340$

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Let me know if you would like further clarification or additional explanations!

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Related Questions:

1. What is the significance of arranging numbers for maximization or minimization in multiplication problems?
2. How would the results change if an additional digit were included (e.g., 9)?
3. Can the strategy used here apply to addition instead of multiplication?
4. How can this problem be solved systematically for larger sets of numbers?
5. How would placing 0 in different positions affect the calculations?

Tip:

When solving optimization problems involving digits, always prioritize placing the largest digits in higher place values (leftmost) for maximization and the smallest digits in higher place values for minimization.