Let x = the no. of 25 cent stamps and 
let y = the no. of 15 cent stamps

25x + 15y = 500

5x + 3y = 100

3 is the smallest coefficient in absolute value.
So write 5 and 100 in terms of their nearest multiple
of 3.  So we write 5 as 6-1 and 100 as 99+1.





Divide every term by 3



Isolate all the fractions on the right:



The left side must be an integer, say A,
so the right side must be equal to 
integer A also, so

 and 

Clear the second of fractions by multiplying
through by 3:



Solve for x:



Substitute in 







Solve for y



So now we have



Now since he bought some stamps,

 and 

So

 and 

 and 

 and 

So since A is an integer 

So there are 6 solutions:

A     x=3A-1   y=35-5A
1       2         30
2       5         25
3       8         20 
4      11         15 
5      14         10
6      17          5 

Edwin