Negative Multiples or Always Stay Positive??

[relentlesspursuito700plus]

I read in a GMAT study guide (Jeff Sackman’s GMAT Math Bible) that multiples and factors are always positive. Is that really true? I am confused because I think he contradicts himself in this book.

Jeff says, “As a side note: Questions will appear to use the phrasing "multiple of", "factor of", and "divisible by" interchangeably. They mean very similar things, but there’s one key difference. Factors and multiples are always positive. The factors of 24 are the positive numbers listed above, for instance. However, 24 is divisible by several negative numbers; 24*6 = integer, so 24 is divisible by -6. It’s not something to wrack your brain trying to understand too thoroughly; it just is.”

Then he gives a DS problem:

295. What is the sum of the numbers in a list of m even integers?
(1) The largest integer on the list is 12.
(2) The list consists of 6 consecutive multiples of 4.

It’s obviously not A. It’s not B either. A & B combined, therefore C, seems like a possibility, but earlier Jeff said that multiples are always positive. So the answer, according to that logic, would be E. But the answer he gives is C.

He explains: “295. C EXPL: To and the sum of the numbers, we need to and both the number of integers, and possibly the pattern that determines which numbers are included. Statement (1) is insufficient: the list could be of any length, and we have no idea what the other numbers are. Statement (2) is also insufficient: while this is helpful, we don’t have any idea how large or small the numbers are. Taken together, the statements are sufficient. Knowing that the list consists of 6 consecutive multiples of 4 and that the largest number is 12, the list must be: (-8;-4; 0; 4; 8; 12) Choice (C) is correct.”

What’s going on here? Am I misinterpreting something?

I noticed some wrong answers in his books, but they were just silly editing errors I’m sure. But this one is a little different. He’s REALLY good by the way. I HIGHLY recommend his site and his materials. Definitely better than most books out there. He may be a bit confusing here, but he really breaks down the GMAT into different topics, which allows you to really concentrate your effort on addressing your weaknesses.

[Ian Stewart]

Your interpretation is correct- multiples do not need to be positive. -8 is certainly a multiple of 4, for example, because it is equal to (-2)*4. So the solution you've posted above to the problem is correct, but the description of the distinction between the various phrases is not.

[lunarpower]

Your interpretation is correct- multiples do not need to be positive. -8 is certainly a multiple of 4, for example, because it is equal to (-2)*4. So the solution you've posted above to the problem is correct, but the description of the distinction between the various phrases is not.

under the traditional mathematical definition, yes, 0 and negative multiples are counted as 'multiples'.

HOWEVER,

to date, as far as i know, there has never been an official problem requiring the use of 0 or negative numbers as 'multiples' of positive integers. in fact, every single problem dealing with factors, multiples, primes, divisibility, and the like has been restricted, by fiat, to positive integers.

if anyone on this thread knows of an OFFICIAL problem that is an exception to this rule - i.e., a problem propounded by gmac themselves that uses 0 or negatives as 'multiples' of some positive integer - please mention that problem.

until that time, though, you can rest assured that you won't have to worry about this little definitional quibble, because the gmat's official problems are restricted to positive multiples/factors/primes/etc.

[erukumk]

Your interpretation is correct- multiples do not need to be positive. -8 is certainly a multiple of 4, for example, because it is equal to (-2)*4. So the solution you've posted above to the problem is correct, but the description of the distinction between the various phrases is not.

under the traditional mathematical definition, yes, 0 and negative multiples are counted as 'multiples'.

HOWEVER,

to date, as far as i know, there has never been an official problem requiring the use of 0 or negative numbers as 'multiples' of positive integers. in fact, every single problem dealing with factors, multiples, primes, divisibility, and the like has been restricted, by fiat, to positive integers.

if anyone on this thread knows of an OFFICIAL problem that is an exception to this rule - i.e., a problem propounded by gmac themselves that uses 0 or negatives as 'multiples' of some positive integer - please mention that problem.

until that time, though, you can rest assured that you won't have to worry about this little definitional quibble, because the gmat's official problems are restricted to positive multiples/factors/primes/etc.

Does that mean, we have to count 0 every time we count the factors of a number ex for number 12

is it 1,2,3,4,6,12
or
0,1,2,3,4,6,12

thanks in advance
mahesh

[Ian Stewart]

Does that mean, we have to count 0 every time we count the factors of a number ex for number 12

is it 1,2,3,4,6,12
or
0,1,2,3,4,6,12

thanks in advance
mahesh

No, zero is absolutely not a factor of 12. Zero is a multiple of 12, not a divisor of 12 -- in fact, zero is not a divisor of any number, because you cannot divide by zero.

[goelmohit2002]

Thanks. Is 0 a multiple of say 4 ?

Can someone please tell what needs to be in DS questions.....

Can someone please tell what will be answer to this question:

What is the value of positive integer N
a) N has only two factors.

Is the statement "a" alone sufficient to answer the same or not ?

Since we do not consider the -ve factors, the above statement is insufficient....as all the prime numbers have two positive factors.

But if we consider both +ve and -ve factor...then only one +ve integer i.e. 1 has only two factors...i.e. 1 and -1...so SUFFICIENT....

[Ian Stewart]

Thanks. Is 0 a multiple of say 4 ?

Can someone please tell what needs to be in DS questions.....

Can someone please tell what will be answer to this question:

What is the value of positive integer N
a) N has only two factors.

Is the statement "a" alone sufficient to answer the same or not ?

Since we do not consider the -ve factors, the above statement is insufficient....as all the prime numbers have two positive factors.

But if we consider both +ve and -ve factor...then only one +ve integer i.e. 1 has only two factors...i.e. 1 and -1...so SUFFICIENT....

Technically, yes, 0 is a multiple of 4, because you can multiply 4 by an integer (0) to get 0. You won't need to worry about this on the GMAT, however; divisibility and multiples questions always discuss positive integers only.

And for that reason, you won't need to worry about a DS question like the one you suggest above. If that were on the real GMAT, Statement 1 would certainly read: "a) N has only two positive factors."

[jo1776]

Your interpretation is correct- multiples do not need to be positive. -8 is certainly a multiple of 4, for example, because it is equal to (-2)*4. So the solution you've posted above to the problem is correct, but the description of the distinction between the various phrases is not.

under the traditional mathematical definition, yes, 0 and negative multiples are counted as 'multiples'.

HOWEVER,

to date, as far as i know, there has never been an official problem requiring the use of 0 or negative numbers as 'multiples' of positive integers. in fact, every single problem dealing with factors, multiples, primes, divisibility, and the like has been restricted, by fiat, to positive integers.

if anyone on this thread knows of an OFFICIAL problem that is an exception to this rule - i.e., a problem propounded by gmac themselves that uses 0 or negatives as 'multiples' of some positive integer - please mention that problem.

until that time, though, you can rest assured that you won't have to worry about this little definitional quibble, because the gmat's official problems are restricted to positive multiples/factors/primes/etc.

==========================================
Hello, this my first post so pardon me if I made a typing error.
There was a question I faced three years ago. It read as follows: Between -100 and 100, how many multiples are there for the number 3?. Answers choices that I remember were 33, 34, 66, 67. I approached it by finding the nearest multiple for 3 just under 100, which is 99. I divided 99 by 3 and I got 33 multiples. Then I did the same thing for the negative numbers dividing -99 by 3 = 33 multiples as well. I rushed for selecting the answer as 66 but when I asked a math professor one day later, he said am totally wrong!! It should be 67 multiples because 0 is also a multiple!!!!!! I will never forget such a deceptive question. It embarks on two conceptual questions: 1-Can multiples include negative numbers? 2-Can zero be a multiple? The test makers knew that many students will fall in such a trap of "definitions" and I was one of them!!
Thanks

[Testluv]

Your interpretation is correct- multiples do not need to be positive. -8 is certainly a multiple of 4, for example, because it is equal to (-2)*4. So the solution you've posted above to the problem is correct, but the description of the distinction between the various phrases is not.

under the traditional mathematical definition, yes, 0 and negative multiples are counted as 'multiples'.

HOWEVER,

to date, as far as i know, there has never been an official problem requiring the use of 0 or negative numbers as 'multiples' of positive integers. in fact, every single problem dealing with factors, multiples, primes, divisibility, and the like has been restricted, by fiat, to positive integers.

if anyone on this thread knows of an OFFICIAL problem that is an exception to this rule - i.e., a problem propounded by gmac themselves that uses 0 or negatives as 'multiples' of some positive integer - please mention that problem.

until that time, though, you can rest assured that you won't have to worry about this little definitional quibble, because the gmat's official problems are restricted to positive multiples/factors/primes/etc.

==========================================
Hello, this my first post so pardon me if I made a typing error.
There was a question I faced three years ago. It read as follows: Between -100 and 100, how many multiples are there for the number 3?. Answers choices that I remember were 33, 34, 66, 67. I approached it by finding the nearest multiple for 3 just under 100, which is 99. I divided 99 by 3 and I got 33 multiples. Then I did the same thing for the negative numbers dividing -99 by 3 = 33 multiples as well. I rushed for selecting the answer as 66 but when I asked a math professor one day later, he said am totally wrong!! It should be 67 multiples because 0 is also a multiple!!!!!! I will never forget such a deceptive question. It embarks on two conceptual questions: 1-Can multiples include negative numbers? 2-Can zero be a multiple? The test makers knew that many students will fall in such a trap of "definitions" and I was one of them!!
Thanks

If the question said "positive" multiples, the answer is 33 because 0 is not positive. (0 is neither positive nor negative).

If the question said "non-negative" multiples, the answer is 34 because 0 is a non-negative (and non-positive) multiple of every integer.

If the question just said "multiples", you have the 33 positive multiples, the 33 negative multiples, and 0 for a total of 67.

I don't know if it's by fiat (although that seems to be the case) BUT the wording of every single GMAT problem I've seen involving these concepts always stipulates "positive" multiples (as Ian points out above: "divisibility and multiples questions always discuss positive integers only".) And, if a GMAT problem ever says "multiple" without the tag "positive", then you would have to include zero and negative multiples, because, on the GMAT, we can't make any assumptions; however, that just won't happen b/c, again, in these kinds of problems, the test-maker always includes the tag "positive" before "multiple".

Long story, made short: in these kinds of problems, simply ignore zero and negative numbers (unless the question fails to include the tag "positive", which has never, to my knowledge, happened thus far).