Application of a Linear Diophantine Equation: Number of Stamps

This video provides an application of a Diophantine equation that is solved using congruence.

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Answer x = 121-8x y = 4+ 5k, maximum stamps 125, least amount of stamps 80

How can you make 5 cents and 8 cents stamps with $6:37?
What is the smallest and largest number of stamps you can use?

Let x =5 cents stamps, and y = 8 cents stamp; hence
5x + 8y = 637 cents
~ will be used for congruent
0x + 3y ~ 2 (mod 5)
3y~ 12(mod 5) add 10
y ~ 4 (mod 5) divide by 3
Hence y= 4 + 5k
Since 5x + 8y = 637 then
5x + 8 (4+5k) = 637 substitute y=4+5k into the equation
5x + 32 + 40k =637 cents
5x = 637-32 - 40k
5x = 605 - 40k
x = 121 - 8k (divide both sides by 5)
So the equations are
y= 4+ 5k and x = 121-8k
When k= 1
y= 4+ 5(1) = 9 and
x = 121 - 8(1) =113
Checking
9 stamps at 8 cents = (8)(9) =72 cents
113 stands at 5 cents = 113 x 5 = 565 cent
565 cents + 72 cents =637 cents
Checking using k=2
y= 4+ 5(2) = 14
x = 121 - 8(2) = 105
14 x 8 = 112 cents
105 x 5= 525 cents
112 + 525 = 637 cents
What's the largest number of stamps. This will occur when there is
the smallest number of 5-cent stamps (since the CHEAPER THE STAMPS the
More stamps you will get) that satisfies the equation, x = 121 - 8k
let x =1, hence 1 = 121-8k
8k = 121-1
8k = 120
k= 15
When k = 15,
y = 4 + 5 (15) since y = 4 + 5k
y = 4 + 75
y = 79 stamps
79 stamps x 8 cents = 632 cents
1 stamp x 5 cents = 5 cents
632 + 5 =637 cents
So the smallest number of stamps = 80 stamps
What's the largest number of stamps
If you could get 1 y-stamp since the y-stamp is the most expensive, you
would have the largest number of stamps. But remember, you don't want to
have changes (cents) to remain.
Using the two equations y = 4 + 5k, and x = 121 -8k
let k= 0, y = 4 + 5(0) and x = 121 - 8(0)
then y=4 and x =121
4 x 8 = 32 cents 121 x 5 = 605 cents
So you would 4 + 121 = 125 stamps
To get the least number of y stamps, k has to be less than 0. But if k= -1
you get y= 4 + 5(-1) = 4-1 = -1. And in this case x = 121- 8(-1) = 129
That would exceed the 637 cents you have to buy the stamps as 129 x 5
= 645 .. Recall that in this, y=-1 or -8 cents. 645- 8 cents = 637 cents.
So the maximum amount of stamps will happen when k = 0,
hence y =4 and x =121
4 + 121 = 125 stamps
So the most amount of stamps you can get with 637 cents is 125 stamps (no change left over and have both 5 cents and 8 cents stamps).
The least amount, as aforementioned, is 80 stamps)