If you have only 5 cent stamps and 3 cent stamps, how many
ways can you put 23 cents postage on a letter?

Let x = the number of 5 cent stamps and y = the 
number of 3 cent stamps

5x + 3y = 23

6x - x + 3y = 24 - 1

2x - x/3 + y = 8 - 1/3

2x + y - 8 = x/3 - 1/3 = A

Note: The right side above must be an integer because
 the
left side is.  Let this integer be A

x/3 - 1/3 = A

x - 1 = 3A

x = 3A + 1

2x + y - 8 = A

2(3A + 1) + y - 8 = A

6A + 2 + y - 8 = A

y = 6 - 5A

Now since x > 0 and y > 0

3A + 1 > 0 and 6 - 5A > 0

A > -1/3  and A < 6/5

-1/3 < A < 6/5

Two integers are possible for A, 0 and 1

For A = 0 and x = 3A + 1 = 3(0) + 1 = 1
             y = 6 - 5A = 6 - 5(0) = 6 

So one solution is 1 5-cent stamp and 6 3-cent
stamps.

For A = 1 and x = 3A + 1 = 3(1) + 1 = 4
             y = 6 - 5A = 6 - 5(1) = 1

So the other solution is 4 5-cent stamps and 
1 3-cent stamp.

Edwin