I am unable to prove the problem described below. I have absolutely no clue as to how this proof should be derived.

The answer is provided in the book, and the question does not really ask that the answer should be proven, but I cannot fathom how the answer can be certain when it's not proven.

Please help me with this if you are able.

**Problem:**Imagine that we only had two different types of coins. One coin type with a value of 4 and one coin type with a value of 7.

Examples of amounts that can be paid with these coins:

- 7 (one coin with a value of 7).
- 8 (two coins with a value of 4).
- 9 is not available.
- 10 is not available.
- 11 (one coin with a value of 4 and one with a value of 7).

**not**pay by using these coins?

__Assumptions__- The question only involves whole numbers (only whole numbered coins are available).
- We have access to an endless amount of whole numbered coins of each value.

__Where I am at:__I am unable to understand if it is possible to prove the highest amount that can not be paid.

My equation so far is:

4x + 7y = z, where x, y and z are whole positive numbers.

I can test different amounts that can not be paid, but as I cannot test all possible combinations of x and y, I am unable to verify the answer with 100% certainty.

Appreciated any help here!