Question 1053002: What are the amounts of postage that can be made with an unlimited supply of 5-cent and 8-cent stamps?
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! Any amount in cents, except
1, 2, 3, 4, 6, 7, 9, 11, 12, 14, 17, 19, 22, or 27.
For larger amounts, there may be multiple ways to make that amount.
You can make up a 40-cent postage with five 8-cent stamps or with eight 5-cent stamps, for example.
You can, of course make any amount that is a multiple of 8 cents or 5 cents.
What about numbers that are not multiples of 5?
Those numbers, divided by 5, will have a remainder of 1, 2, 3, or 4.
If one such number is large enough, you can make the difference by replacing some 5 cent stamps with some 8 cent stamps.
Using one 8-cent stamp to replace  5-cent stamp, you increase the total amount by  cents.
Using two 8-cent stamps to replace  5-cent stamps, you increase the total amount by  cents.
Using three 8-cent stamps to replace  5-cent stamps, you increase the total amount by  cents.
Using four 8-cent stamps to replace  5-cent stamps, you increase the total amount by  cents.
So, if the amount is  or more,
it does not matter if the amount is divisible by 5,
because any remainder from dividing the amount by 5 can be made up by replacing some 5-cent stamps with 8-cent stamps.
For example, if you want  cents worth of postage,
knowing that  5-cent stamps would make  cents,
as explained above, you can increase that amount by  cents
by replacing 5-cent stamps with 8-cent stamps.
So  5-cent stamps and 8-cent stamps make an amount , in cents, of
 .
That is not the only way to make that amount,
seven 8-cent stamps and one 5-cent stamp works too,
but starting from a multiple of 5 gives you a way that works, and is easy to understand.
NOTE:
If you have been learning about it in math class, you could say that the problem relates to "clock arithmetic" or modular arithmetic."
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