At East End Mall, burgers can be bought in quantities of either 6,9 or 20 only. For example one can purchase `26(20 + 6), 15(6 + 9) and 24(9 xx 2 + 6)` etc. but can't purchase 10 burgers i.e., without any lot. What is largest number of burgers that can be purchased ?

Options

A. a. 83
B. b. 37
C. c. can't be determined
D. d. none of (a), (b) and (c)

Answer

To solve the problem of finding the largest number of burgers that cannot be purchased using quantities of 6, 9, or 20, we can follow these steps:

### Step 1: Identify the quantities available
The quantities of burgers that can be purchased are 6, 9, and 20.

### Step 2: Find the GCD
We need to find the greatest common divisor (GCD) of the numbers 6, 9, and 20.
- The GCD of 6 and 9 is 3.
- The GCD of 3 and 20 is 1.
Thus, the GCD of 6, 9, and 20 is 1.

### Step 3: Understand the implications of the GCD
Since the GCD is 1, it means that we can generate a wide range of numbers using combinations of 6, 9, and 20. However, there will still be some numbers that cannot be formed.

### Step 4: Use the Chicken McNugget Theorem
For two coprime integers \( m \) and \( n \), the largest integer that cannot be formed using combinations of \( m \) and \( n \) is given by \( mn - m - n \). However, since we have three numbers, we will need to analyze combinations.

### Step 5: Analyze combinations of 6 and 9
The combinations of 6 and 9 can generate all multiples of 3 (since both are multiples of 3). The largest number that cannot be formed using just 6 and 9 is 3 (as we cannot form 1, 2, or 3 using just 6 and 9).

### Step 6: Include 20 in the combinations
Adding 20 allows us to generate more numbers. We can form any number that is greater than or equal to 20 plus any multiple of 3.

### Step 7: Identify numbers that cannot be formed
To find the largest number that cannot be formed, we need to check numbers below 20:
- 1, 2, 3, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19 are all numbers that cannot be formed using combinations of 6, 9, and 20.
- The largest of these is 43.

### Conclusion
The largest number of burgers that cannot be purchased using the quantities of 6, 9, or 20 is **43**.

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