An eccentric casino owner decides that his casino should only use chips in `$5` and `$7` denominations. Which of the following amounts cannot be paid out using these chips?

Options

A. `$31`
B. `$29`
C. `$26`
D. `$21`

Answer

To determine which amounts cannot be paid out using only $5 and $7 chips, we will analyze each option one by one. The goal is to express each amount as a combination of the two denominations.

### Step-by-Step Solution:

1. **Check $31:**
- We need to express $31 as a combination of $5 and $7.
- Let's try:
\[ 31 = 7 \times 3 + 5 \times 2 \]
- Calculation:
- \(7 \times 3 = 21\)
- \(5 \times 2 = 10\)
- \(21 + 10 = 31\)
- Therefore, $31 can be paid out using $5 and $7 chips.

2. **Check $29:**
- Now, we check $29:
- Let's try:
\[ 29 = 7 \times 2 + 5 \times 3 \]
- Calculation:
- \(7 \times 2 = 14\)
- \(5 \times 3 = 15\)
- \(14 + 15 = 29\)
- Therefore, $29 can also be paid out using $5 and $7 chips.

3. **Check $26:**
- Next, we check $26:
- Let's try:
\[ 26 = 7 \times 3 + 5 \times 1 \]
- Calculation:
- \(7 \times 3 = 21\)
- \(5 \times 1 = 5\)
- \(21 + 5 = 26\)
- Therefore, $26 can also be paid out using $5 and $7 chips.

4. **Check $21:**
- Finally, we check $21:
- Let's try:
\[ 21 = 7 \times 3 + 5 \times 0 \]
- Calculation:
- \(7 \times 3 = 21\)
- \(5 \times 0 = 0\)
- \(21 + 0 = 21\)
- Here, $21 can only be expressed using $7 chips and does not include any $5 chips.

### Conclusion:
- The amount that **cannot** be paid out using $5 and $7 chips is **$21**.

### Final Answer:
**The amount that cannot be paid out is $21.**

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