Here's some hints:

• 3.23: This is fairly similar to Example 3.3, so start by making sure you understand that. Note: What does the prime factorization of a perfect cube look like? Since 120^6 is a perfect cube itself the perfect cube factors still come in nice pairs.
• 3.26: If the store can give change, it's possible to pay for any amount of the form 3*K + 11*L for integers (positive or negative) K and L. What K and L work to get 1? (This is guaranteed by Bezout's). If no change can be given, note K and L must be non-negative. This is a little more complicated. Hint: Once you can pay for X dollars, note adding 3 dollar bills allows you to pay for X+3, X+6, etc.
• 3.30: Is there a particular step you're confused about here? The proofs can be fairly tricky sometimes, but it's important to work through it piece by piece slowly.
• 4.10: Hint 1: Start by thinking about what ones digit is possible. Hint 2: Note if our number is 10x + y, then $$(10x+y)^2 \equiv 20xy + y^2 \pmod{100}.$$Consider the different ones digits (y) from here.
• 4.25: If the notation is confusing here, start by just considering a fixed k. For example, try to do the proof for $k=5$, so the number can be written in the form $a\cdot 10^5 + b\cdot 10^4 + c\cdot 10^3 + d\cdot 10^2 + e\cdot 10 + f$.
• 4.26: Using 4.25b, note we can calculate $52\cdots 2$ mod 11 based on how many 2s there are (using the alternating sum of the digits).

Hope this helps! Note I saw you started a submission for this chapter 4 homework, which is due on Thursday, good job! Note once the submission closes on Thursday, you'll be able to review your submission and see the solution ideas for all the homework problems. Note: I also submitted a blank Chapter 3 Assignment for you, so you can click on that assignment now and see solution ideas for the chapter 3 problems as well.