I’m taking the GRE prep course and the prof. puts this sample question on the overhead projector:
If you add the two lowest factors of the intergers 8 and 6, you would get:
It turns out that -14 is the correct answer. WTF?? I was always taught that factors of various intergers are WHOLE, POSTIVE numbers. No one EVER said that negative numbers could be used as factors.
When I raised in an objection in class, the prof. said that this question was once on a GRE test and several students had objected and took the case to court. Expert testimony was given and that, yes, indeed, negative intergers can be factors. This seems to go against everything I was taught, or at least, it was never mentioned in any math classes I took.
Now, I realize the GRE makes you stretch your mind and has you look at things in a very different light, yet this seems to be really pushing the boundries.
Can any of the mathematicians or number theorists out there confirm or deny this?
But I’m going to try anyway.
Well, the factors of 6 are 1, 2, 3, 6. However, when you multiply a negative number with another negative number, you get a positive. So couldn’t the factors also be -1,-2,-3,-6?
I guess that would mean the factors could be ± 1,2,3,6, and we’re all just lazy so we don’t add the “-” symbol.
And that would also be ± 1,2,4,8 for 8. So -6±8 is -14.
My understanding is that factors are numbers that are multiplied to form a product. -8 and -1 are factors of the product 8, but so are 16 and 0.5. I agree with you that the question should be worded more precisely to show that they mean integral factors.
Unless your prof. has a citation, I’d be extremely dubious. The mathematical terms used in GRE questions are defined in the materials, and I’m pretty sure that that answer is not consistent with their definitions. I’ll dig 'em out, just to make sure.
I would probably have answered C) 2. But the question did say integers implying that negative integers could be considered. I agree with Squink the question is UGLY.
If you want to define primes over the integers or the complex integers, you must first define a unit as a factor of one. Integral units are 1 and -1. Complex integral units are 1, i, -1, and -i. A non-zero non-unit number p is prime if whenever p = a b then a or b is a unit. Integral primes are those integers that have exactly four divisiors. Complex integral primes have exactly eight divisors.
Actually, that’s the definition of an irreducible. A prime p is usually defined as: Whenever p divides ab, then p divides a or p divides b.
Of course, to talk about factors, irreducibles, and primes and so forth, you have to be clear on what set you’re talking about–the natural numbers, integers, rationals, reals, complex, or whatever. So, for example, 3 doesn’t divide 7 in the integers, but it does in the rationals. For that reason I do think it’s a misleading question.
I agree, this seems to be a back-handed way of tripping you up. While no one has ever said that factors couldn’t be negative numbers, it was always heavily implied that you used only positive number. None of my math teachers ever said that negative numbers were factors; tho I do agree with PepperlandGirl, when you multiply two negatives, you get a positive.
Dr. Matrix, I’m with you, too! No a cite wasn’t given, but the story about the lawsuit came from the prof, too.
I know GRE test is supposed to make you think and look at things in different ways, but this seems like a very sneaky and underhanded way of doing it. I’ve been thinking of this and it seems that they’ve changed the definition of what is “factoring” for this question or used a definition that’s never been taught in schools, which I think is underhanded and plain wrong.
There is at least one instance where negative factors are quite important: the rational root thrm… If they taught this in your high school algebra class, your teacher would have to mention negative factors. If not, I can’t picture them them showing up (undoubtedly there are other applications, just none that I recall having seen).
Full disclosure: I would have answered 2 (actually 4, because every time I read factors, I think prime factors. But that just me).