The people of Freedonia use two coins one worth 4 and one wo

The people of Freedonia use two coins, one worth 4$ and one worth 10$. The possible amounts made are 8$, 10$, 12$, ...

Prove the possible amounts made are subsequent even numbers after 8$.

Solution

There are two types of even numberes beyond 8$

4n type

And 4n+2 type

4n types are all multiples of 4 of can be made with n 4$ coins

First 4n+2 type after 8 is 10

For any 4n+2>=10 can be made with one 10$ coin and m 4$ coins

where m = (4n+2-10)/4