When sum of remainders becomes divisible

  1. Hi. I need to sort out some concepts and terminology. I was wondering if there are algorithms and terminology surrounding the following situation.

    Lets say I want to buy some object for an amount of money, but the object cost less than the amount of money I have. I will have a remainder of money. Lets say that this purchase on it's own is inefficient, but, once I make more money, keep coming back and buying the same object, eventually I will be able to buy a final object at just with the accumulated remainders of money. I simply delayed making an efficient use of my money till later.

    I know that I need just solve for 'x' to find out how many purchases it takes to use the saved up money in:

    x*remainder = object price

    Then I can also find out how much non-remainder money I had to burn to begin with in order to get the "freebie".

    Sooooo, my question is if anyone knows some terminology that explains this scenario or can name concepts in science or finance ...etc. that they see this analysis being used. I simply like putting words onto my math :)

    Thank you.
  2. jcsd
  3. You didn't say whether or not you keep buying the object with the same amount (if you don't, the remainder will be different, so this needs clarification). I will assume you keep buying the object with the same payment for the sake of providing some sort of answer.

    If c is the object cost, and p is your payment, then the remainder of this transaction can be expressed as: p mod c

    So then, we want to know the number of times we need to accumulate remainders to pay for the object with collected remainders. Let this number be k.

    k[p mod c] >= c

    You used an equals sign, but note that k is a natural number. So, if you could pay for the item exactly from remainders, then c / [p mod c] has to be a natural number. That won't always be possible.

    So basically, what I think the name / topic you are looking for is Modular Arithmetic
    1 person likes this.
  4. A couple of things.

    1. Is the initial value the same as the person's salary? This will change the equations noted in the previous posts.

    2. I claim there always is a solution. This depends on the treatment of p as a natural number. Certainly, $1.99 doesn't look like a natural number because of the decimal. But if you convert to cents it is just 199.

    There are various strategies. In terms of number theory this can be solved as a simple Diophantine equation. The basis of this is modular arithmetic.

    Another strategy is called linear programming. This would involve recursive equations.

    I think this problem would be different depending on whether you are approaching it from a pure math background or a financial. If financial, I think there is a question about whether a remainder of a penny or a few cents is significant.
    1 person likes this.
  5. Thanks guys. It seems my question may have been a bit vague, but I DID get a good starting point in the replies for more investigation. Thanks all.