Let's break down the first three problems from the image:

1. Arrange the digits 1, 2, 3, 4, and 5 to make a two-digit and a three-digit number. Use a calculator to multiply the numbers. Try several different arrangements of the digits to determine the arrangement that results in the greatest possible product.

Explanation: To maximize the product, you should place the largest digits in the highest place values.

Try different combinations, but the best approach will be to make the three-digit number as large as possible while keeping the two-digit number reasonably large as well.

Example:

• One arrangement: $542 \times 31 = 16,802$
• Another: $432 \times 51 = 22,032$
• Best arrangement found: $543 \times 21 = 22,803$

Thus, the greatest product is $\mathbf{543 \times 21 = 22,803}$.

2. Repeat the problem and arrange the digits so that you obtain the least possible product.

Explanation: To minimize the product, reverse the strategy by putting the smallest digits in the highest place values.

Example:

• One arrangement: $123 \times 45 = 5,535$
• Another: $134 \times 25 = 3,350$
• Best arrangement found: $123 \times 45 = 5,535$

Thus, the least possible product is $\mathbf{123 \times 45 = 5,535}$.

3. Analyze the results in Exercises 1 and 2. Given any five non-zero digits, what placement of the digits in a two-digit and a three-digit number will guarantee the greatest product and the least product?

Explanation: For the greatest product:

• Place the largest available digits in the hundreds and tens places for the three-digit number and the tens place for the two-digit number, as this maximizes their value.

For the least product:

• Place the smallest available digits in the highest place values to minimize the value of the numbers being multiplied.

In summary:

• Greatest product: Put the largest digits in the highest places of both numbers.
• Least product: Put the smallest digits in the highest places of both numbers.

Do you have any questions or would you like further details on any step?

Related Questions:

1. How does place value influence the result of multiplication?
2. What strategy can be used to minimize the product for any set of digits?
3. Can you find another combination of five digits that results in a higher product than the one listed?
4. Why is it important to consider the arrangement of digits when multiplying two numbers?
5. What would happen if you tried this exercise with six digits?

Tip:

When working with digits to maximize or minimize products, focus on the place value—the higher the digit in the larger place value, the greater its impact on the final product.