is this response using proof by induction "Alright, let's tackle this with a bit of algebra and logical reasoning. We want to show that any amount of money, let's call it M , greater than 8 dollars, can be made using just 3-dollar and 5-dollar bills.

First, let's establish a foundation with some specific examples to see the pattern:

For $8: We can't make $8 exactly with just 3s and 5s, so this is our starting point.

For $9: We can use three 3-dollar bills (3x3).

For $10: We can use two 5-dollar bills (2x5).

For $11: We can't make $11 exactly with just 3s and 5s.

For $12: We can use four 3-dollar bills (4x3).

Now, notice something interesting:

Once we have $12 (which is 3x4), we can make any amount above by adding more 3-dollar bills.

But what about the numbers we skipped, like $11? Well, we don't need to worry about them because we're looking for any amount greater than $8, and we've found a way to make $9, $10, and $12. From $12 onwards, we can add 3-dollar bills to cover all subsequent amounts.

To put this into a general form, we can express amounts using 3-dollar and 5-dollar bills as follows:

M = 3a + 5b

where M is the total amount, a is the number of 3-dollar bills, and b is the number of 5-dollar bills.

We've shown that for M = 9 and M = 10 , we have solutions. For any M > 10 , we can use the combinations we found for $9 and $10 and just add enough 3-dollar bills to reach M . Specifically, for any M > 10 , we can write:

M = 9 + 3n or M = 10 + 3n

for some non-negative integer n . Here, n represents the additional number of 3-dollar bills we can add to our base cases of $9 and $10 to reach any desired amount M .

This shows that using a combination of 3-dollar and 5-dollar bills, we can create any amount greater than $8."