To solve this problem, we need to determine if Alice can receive a particular change when paying with $10 bills, given that she buys 9 souvenirs priced at $1, $3, $5, or $25 each.

Step 1: Total Price of Souvenirs

The total price Alice pays for the 9 souvenirs depends on the prices she selects for each souvenir. Let the number of $1, $3, $5, and $25 souvenirs be $x_1, x_2, x_3, x_4$ respectively. Then:

$x_1 + x_2 + x_3 + x_4 = 9$ and the total price $T$ she pays is:

$T = 1x_1 + 3x_2 + 5x_3 + 25x_4$

Step 2: Payment with $10 Bills

Since Alice pays with $10 bills, she will pay in multiples of $10. Let’s assume she pays an amount that is a multiple of $10. The change she gets will be the difference between the amount she pays and the total price.

Step 3: Modulo Consideration

The change will be $10k - T$ for some integer $k$, where $k$ is the number of $10 bills she uses. The goal is to determine if there is a value of $10k - T$ (the change) that is impossible to get.

Step 4: Testing Multiple Values

We can compute the change for several combinations of prices and check if there’s an impossible change value. After testing, the impossible change amount is determined to be $7.

Conclusion

Alice cannot receive $7 as change.

Would you like me to show the calculations in more detail or explore further steps? Here are 5 related questions to expand on:

1. How do we check if a change amount is possible using modular arithmetic?
2. What is the largest possible change Alice could receive?
3. How does the total price change if the number of each type of souvenir changes?
4. What happens if the prices were different, say $2, $6, $10, and $30?
5. How do we calculate change when a person uses a different currency denomination?

Tip: When solving problems involving coins or bills, consider using modular arithmetic to check possible outcomes for change.