SOLUTION: People be confusing me with this Cookies are sold singly or in packages of 13 or 39. With this packaging, how many ways can you buy 78 cookies?

Just by trial and error you can easily get the answer

1. 2 boxes of 39, 
2. 1 box of 39 and 2 boxes of 13, and
3. 6 boxes of 13.

If you're supposed to use algebra or number theory, then

Let x = the number of packages of 13 cookies
Let y = the number of packages of 39 cookies.

That leads to the Diophantine equation:

13x + 39y = 78

Divide through by 13

x + 3y = 6

x = 6 - 3y

x ≥ 0, y ≥ 0

6 - 3y ≥ 0

   -3y ≥ -6
     
     y ≤ 2

Also y ≥ 0 so

0 ≤ y ≤ 2

Since y is an integer, y can only take on the values 0,1, and 2.

So the answer is 3.

They are 2 boxes of 39, 
1 box of 39 and 2 boxes of 13, and
6 boxes of 13. 

Edwin