**Here’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.

*Rob 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.*