An AB Production (Odds and Evens)

[poojakrishnamurthy1]

Hi,

I encountered this question in CAT 1.

Question -

Is the product of a and b odd?

(1) a has only 2 factors
(2) b = 2x + 1, where x is an integer
My doubt:

The Original Answer for this question is E but I answered C. The explaination given is that since a has just 2 factors, it must be a prime number. However, the explaination fails to take into account negative factors. Because statement (1) reads "a has only 2 factors" and it is nowhere mentioned that a has only positive factors, it means that a has a total (including negative and positive factors) of 2 factors. A prime number typically has 4 factors, 2 positive and 2 negative. For eg, 2 has four factors- -2,-1,1, and 2. The only numbers that have 2 factors are -1 and 1, both having 1 and -1 as the two factors.

Ofcourse (1) and (2) by themselfves are insufficient. However, when you combine you know that a can take a value of either 1 or -1; in both cases a is odd. (2) tells us that b is odd. The product of two odd numbers is always odd; therefore, ab must be odd. The answer should be C.

Please advise.

[gmatblr]

The prime factorization of a number "n" is the
unique set of positive prime numbers which, when multiplied together,
yield the number n. And when we list "the factors of" a number, we
only list positive integer divisors. The "factors" of 6 are 1, 2, 3,
and 6; we don't include -1, -2, -3, or -6 in the list.

Hence do not consider negative number under fatcors .

1. a has only 2 factors , implies ....its a prome number ...2,3,5,7 (Note 2 is the only even prime number)
2. b=2x +1 ,indicates b is an odd integer

a can be even /odd
b is odd integer

even + odd = odd
odd + odd = even

Hence E

[poojakrishnamurthy1]

I don't think that when we list the factors of an integer, we only list positive factors. For example if the question stem says that x is a positive number, we can only infer that x>0. We cannot assume that x must also be an integer. X could very well be a fraction also. One thing we should always know on the GMAT is that unless otherwise stated, we cannot assume that the factors of an integer are just positive.

The definiton of a factor as defined by OG 11 (Pag 108) is -

"If x and y are integers and x is not equal to 0, then x is a divisor (factor) of y provided that y=xn for some integer n. In this case y is also said to be divisible by x or to be a multiple of x."

No where in the definition is it mentioned that x and y must be positive. It also means that an integer also has as many negative as positive factors.

The definition of a prime number as defined in OG 11 (page 108) is -

"A prime number is a positive integer that has exactly two different positive divisors, 1 and itself."

Since the definition of a prime number clearly says that a prime number has two different postive factors, one can clearly infer that it must definitely have two different negative factors too.

Coming back to the question, if the stem says that a has 2 factors, it must include all factors, including positve factors. If a has to be a prime number, the stem must say that either "a has 2 positive factors" or "a has four factors".

I still think that the answer should be C.

[RonPurewal]

I don't think that when we list the factors of an integer, we only list positive factors. For example if the question stem says that x is a positive number, we can only infer that x>0. We cannot assume that x must also be an integer. X could very well be a fraction also. One thing we should always know on the GMAT is that unless otherwise stated, we cannot assume that the factors of an integer are just positive.

The definiton of a factor as defined by OG 11 (Pag 108) is -

"If x and y are integers and x is not equal to 0, then x is a divisor (factor) of y provided that y=xn for some integer n. In this case y is also said to be divisible by x or to be a multiple of x."

No where in the definition is it mentioned that x and y must be positive. It also means that an integer also has as many negative as positive factors.

The definition of a prime number as defined in OG 11 (page 108) is -

"A prime number is a positive integer that has exactly two different positive divisors, 1 and itself."

Since the definition of a prime number clearly says that a prime number has two different postive factors, one can clearly infer that it must definitely have two different negative factors too.

Coming back to the question, if the stem says that a has 2 factors, it must include all factors, including positve factors. If a has to be a prime number, the stem must say that either "a has 2 positive factors" or "a has four factors".

I still think that the answer should be C.

this is the subject of some debate, but let's remember that, in such matters, there's exactly one opinion that matters: that of the gmat itself.
and i can tell you this:
the gmat has never, ever, required the use of "negative factors" on an official question.
they just haven't.
and there's no good reason to suspect that they will.
you are right that there have occasionally been questions containing the phrase "positive factor(s)", but that's redundant for our purposes.

in any case, you do have a point, so we should probably just alter the wording of this problem so that statement (1) says "positive factors" - in line with both the gmat's intent (i.e., they don't use negative factors) and its format (i.e., they use very precise wording).

thanks for pointing this out.