At McDonald’s, you can buy chicken strips in a 3 strip box or a 5 strip box. Prove that you can buy exactly n chicken strips from McDonald’s for any n ≥ 8.

To prove that you can buy exactly \( n \) chicken strips from McDonald’s for any \( n \geq 8 \) using only 3-strip and 5-strip boxes, we will use the concept of the **Chicken McNugget theorem** (a variant of the Frobenius coin problem) and induction.

### Step 1: Base Cases
Let's verify that we can buy exactly \( n \) strips for some initial values of \( n \geq 8 \):

1. For \( n = 8 \): Use one 3-strip box and one 5-strip box.  
   \( 8 = 3 + 5 \).

2. For \( n = 9 \): Use three 3-strip boxes.  
   \( 9 = 3 + 3 + 3 \).

3. For \( n = 10 \): Use two 5-strip boxes.  
   \( 10 = 5 + 5 \).

### Step 2: Inductive Step
Assume that you can buy exactly \( n \) chicken strips for some \( n \geq 8 \). Now, let's prove that you can also buy \( n + 1 \) strips.

**Inductive Hypothesis**: Assume that we can form \( n \) strips for some \( n \geq 8 \).

Now consider \( n + 1 \):
- If \( n \) is formed using some combination of 3-strip and 5-strip boxes, you have either at least one 3-strip box or at least one 5-strip box in the current combination.
  
If you have at least one 3-strip box:
- Remove one 3-strip box and add one 5-strip box:  
  This changes \( n \) strips to \( n - 3 + 5 = n + 2 \). 

Since we know \( n \) can be made, and we've shown that \( n + 2 \) can be made, to form \( n + 1 \), we simply use the construct for \( n \) and then just replace a \( n \) case with \( n + 2 \).

Additionally, because our base sequence from \( n = 8, 9, 10 \) covers sequential from there, the above pattern extends upward since each successive can be constructed.

### Proof by Induction
Thus, by induction, since we can construct 8, 9, and 10 as shown and every step upward is attainable via the above changes:

**Therefore, you can buy exactly \( n \) chicken strips for any \( n \geq 8 \).**

Would you like further clarifications or have additional queries? Here are some related questions:

1. What is the smallest number of strips that cannot be purchased using only 3-strip and 5-strip boxes?
2. How does the Chicken McNugget theorem apply to other combinations of box sizes?
3. Can you prove the theorem using a different method, like modular arithmetic?
4. What if McDonald’s introduced a 2-strip box? How would the solution change?
5. Can we generalize this proof to other box sizes beyond 3 and 5 strips?

**Tip:** For problems like this, use base verification and induction combined to confidently prove upward into more generalized cases!

Explore how you can buy exactly n chicken strips from McDonald’s for any n ≥ 8 using 3-strip and 5-strip boxes. Learn the application of the Chicken McNugget theorem and induction in number theory.

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