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Apples and Remainders

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ApplesHere’s a classic question:  

Out of the 6000 apples we harvested, every third apple was too small, every fourth apple was spotted, and every tenth apple was bruised. The remaining apples were good. How many good apples were there?

What is the purpose of constructing a viable mathematical argument or proof? Many people would say that the purpose is to convince another person or yourself that the position being taken is true, beyond a shadow of a doubt. But I think we have focused too much on conviction as the purpose of mathematical argumentation. I think the primary benefit of constructing a viable argument is that it uncovers the way mathematical ideas relate to one another. When you make a viable argument for a claim, it reveals what mathematics the claim relies on. Here is an example:

Students usually start by saying, “Subtract from 6000 the 2000 apples that are too small, the 1500 that are spotted, and the 600 that are bruised to get 1900 good ones left.” This explanation isn’t correct, because the students have subtracted some of the apples twice (e.g., the ones that are spotted and bruised).

After realizing this, students sometimes take a different tack. They might list the first, say, 100 apples and find out the good ones by brute force (there are 47, I think). Then they say that if there are 47 in the first 100, there will be 47 x 60 in the first 6000. But this argument is not sound because it assumes that the second 100 apples (101-200) behave exactly like the first 100, which they don’t.

But if this is done with the first 60 apples instead of the first 100, it does work because every subsequent batch of 60 in fact does behave like the first 60. Why is this?

Good numbers here are precisely the ones with a nonzero remainder when you divide them by 3, 4, and 10. But if any number n is good, then n + 60 will be good, too. The reason is that 60 has no remainder when divided by 3, 4, or 10. This means that when you divide n + 60 by 3, 4, or 10, you get the same remainder as when you divide n by 3, 4, or 10 because the 60 contributes nothing to the remainder. This is the same for n + 120, n + 180, etc. This means that apple n + 60, n + 120, etc., are all good precisely when n is good.

Notice the benefit of pushing through to the viable argument here in that it uncovers an important piece of mathematics that our claim relies on: If I take a number n and divide it by some other number p, I will get the same remainder if I divide n + m by p, provided m is divisible by p. This is a big rule in number theory and is the basis for modular arithmetic.

This mathematical idea doesn’t become apparent if you just stop at the answer or even just stop at how the answer was arrived at.

RobElyRob Ely is an associate professor in the Department of Mathematics at the University of Idaho.  He likes researching student reasoning and how it relates to historical reasoning, particularly with the infinite, infinitesimal, and algebra.  He likes playing hammered dulcimer, bridge, and fetch. 

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