a,b are consecutive integers

so , b= a+1

and ab = a(a+1) = a^2 + a =30x

from 1 : a^2 is div by 21,

Let a=21, so (a^2 + 21 or a) => is always divided by 21

moreover (a^2 + a)/21 =(21^2 + 21)/21 = 22

so we have 22 = 30 x,

i.e x = 22/30 =11/15 = non integer.

1) alone is sufficient

now a = b-1 , so ab = ( b-1)b = b^2 -b

now let b=35. so b^2 is always div by 35 and also by (35^2 -35) or (b^2-b)

so (35^2 -35)/35 = 34

34 = 30 x

x= 30/34 = 15/17 = non integer.

2) alone is also sufficient.


I am getting answer as D: each stem is sufficient to answer the question.

I think I am missing here something. plz help out to why it is C (both stems are required to answer)

Thanks for posting this question abhi

hrishi19884 wrote:a
I am getting answer as D: each stem is sufficient to answer the question.

I think I am missing here something. plz help out to why it is C (both stems are required to answer)

Thanks for posting this question abhi

You forgot that in Data sufficiency,we check whether given data are sufficient enough to reach a conclusion.So,in this question though you have found one value of 'x' suiting the condition,you didn't bothered to check if it is a unique value and if there aren't other values breaching the condition.That's where you have erred in reaching a conclusion.
a^2 is div by 21 means both a and a^2 are divisible by 21(for an integer a).Hence a=21k and a^2=21^2k^2
Eventually,using your steps further
22y=30 x
For y=30 x=22 an integer.....

Yeah, I did bother to find the other values but didn't get those which (a^2 + a) is not div by 21 but a^2 is div by 21

similarly was the case with 35

can you tell me what values they could be?

abhi332 wrote:If a and b are consecutive positive integers, and ab = 30x is x a non-integer?
(1) a^2 is divisible by 21
(2) 35 is a factor of b^2

[spoiler]OA:C
[/spoiler]

This is a bad question. I'll explain why at the bottom of my post, but first:

ab = 30x; is x a non-integer?

Does ab/30 = a (non) integer?

In order for the product ab to be divisible by 30, the product ab has to contain 2, 3, and 5 in its prime factorization. Because they are consecutive, we know exactly one of a and b is even; so, we know ab already has 2 in its prime factorization. The question becomes: does ab have 3 and 5 in its prime factorization?

(1) a^2 is divisible by 21;

In order for a^2/21 to be an integer, a has to have 3 and 7 in its prime factorization. But either of a or b may or may not be a multiple of 5...insufficient.

(2) 35 is a factor of b^2;

therefore, b has 5 and 7 in its prime factorization. But either of a or b may or may not be a multiple of 3...insufficient.

Together, you know that ab is a multiple of 3 and 5. Sufficient in combination.

HOWEVER, this question is flawed in design, as I remarked above. The information in the question stem contradicts the information in the combined statements: The question stem tells us that a and b are consecutive positive integers while the combined statements tell us that both a and b are multiples of 7. But two consecutive integers cannot be multiples of the same positive integer (unless that integer is 1). Without the word "consecutive", the question would be fine.

What is the source of this question?

[spoiler]Source: Manhattan Challenge question[/spoiler]

Testluv wrote:

abhi332 wrote:If a and b are consecutive positive integers, and ab = 30x is x a non-integer?
(1) a^2 is divisible by 21
(2) 35 is a factor of b^2

[spoiler]OA:C
[/spoiler]

This is a bad question. I'll explain why at the bottom of my post, but first:

ab = 30x; is x a non-integer?

Does ab/30 = a (non) integer?

In order for the product ab to be divisible by 30, the product ab has to contain 2, 3, and 5 in its prime factorization. Because they are consecutive, we know exactly one of a and b is even; so, we know ab already has 2 in its prime factorization. The question becomes: does ab have 3 and 5 in its prime factorization?

AMAZING explanation Deepak! Thanks a ton.

I have just one more query.

In the above, you said that ab should contain 2,3,5 in its prime factorization.

So if I consider a = 21 (which has both factors 3 and 7 as per (1))

and if I consider b = 35(which has both factors 5 and 7 as per (2))

so a*b will just be 21*35 and here we haven't accounted "2" anywhere which is needed. Also they a, b aren't consecutive.

So if I take a PRODUCT a*b as 21*35*2 , how we can be so sure that the product contains two consecutive integers(a,b).

and that ab/30 will be a non integer.

It is also possible that a*b is non consecutive(since consecutive a,b do not exists) but it satisfies(1) and (2)

We have no information about what product a*b can be : it can be 21*35*8 as well right?

Hi Hrishi,

ignoring the word "consecutive" in the question stem, and considering the statements in combination would mean that ab has 3, 5 and two 7s in its prime factorization, which is not enough to know whether it is divisible by 30, and the answer would be E.

In data sufficiency, we always have to look at the information in each statement in conjunction with the information in the question stem. Here, the question stem demands that a and b be consecutive positive integers. But, as I explained in my first post, the combined statements make that impossible.

I hope that clears things up.

Thanks Deepak! It is an ambiguous question. I don't think this one is tested in GMAT.