# I don't know under which category or name of this math comes...

#### aldrin

##### New member
Hi,
first of all, I am very bad at numbers. here is the calculation. it looks simple to me but I struggle more to solve this.

I am a stamp vendor so I do sale stamps according to my government provided denomination with certain rules.

stamp paper available denominations are 10,50,100,500,1000,5000,10000,15000,20000,25000

for example if a person wants 17000 then I have provide 7 or 13 sheets using above denomination if possible.

17000 in 7 sheets
number of possiblities

i)

5000 * 3 = 15000
500 * 4 = 2000
_________________________
7 17000
_________________________
so 7 stamp papers total of 17000 amount
At some situation I don't have 5000 denomination at that moment I have to figure out other possibility

ii)

13 sheet for 30000

10000 * 1 = 10000
5000 * 2 = 10000
1000 * 10 = 10000
___________________________
13 30000
so 13 stamp papers total of 30000 amount

This above two example took me 15 mins to figure out.

Is there any formula or method to figure out the denominations in a quick way?

Thanks

#### Jomo

##### Elite Member
You are confusing yourself. 1st you say that you have two ways to sell 17,000 with 7 or 13 sheets.
You then show how you found a way to combine 7 sheets for a total of 17,000.
Then you went on to show how you can sell 13 sheets but magically the total became 30,000. What happened?

#### aldrin

##### New member
@Jomo Thanks for replying, I have to sell the no of sheets by customer's request either 7 or 13 or whatever the number of sheets if only possible.

As you asked for 13 sheets with 17000

5000 1 5000
1000 12 12000
___________________
13 17000

like this I have to quickly calculate this calculation. any method or I don't know what to call this in mathematics.

so the amount may vary from customers as well as the no of sheet. so both are variables (amount and no of sheet).

#### HallsofIvy

##### Elite Member
Those look like "Diophantine equations".
7x+ 13y= 1700 where x and y are positive integers.
First look at 7x+ 13y= 1.
7 divides into 13 once with remainder 6: 7(-1)+ 13(1)= 6.
6 divides into 7 once with remainder 1: 7(1)+ 6(-1)= 1
Replace that "6" with "7(-1)+ 13(1)" to get
7(1)+ (7(-1)+ 13(1))(-1)= 7(1)+ 7(1)+13(-1)= 7(2)+ 13(-1)= 1.
Multiply by 1700:
7(3400)+ 13(-1700)= 1700.
x= 3400, y= -1700 is one solution.
But then x= 3400- 13n, y= -1700+ 7n is also a solution for any n:
7(3400- 13n)+ 13(-1700+ 7n)= 1700- 7(13)n+ 13(7)n= 1700.

Since we want y to be positive y= -1700+ 7n> 0 requires that n be at least 243.
y= -1700+ 7(243)= -1700+ 1701= 1.
With n= 243, x= 3400- 13(243)= 3400- 3159= 241 and subtracting another 243 would make x negative so x= 241, y= 1 is the only positive integer solution.