MGMAt CAT Question

[san2009]

If a and b are positive integers such that
a div by b = 2.86,
which of the following must be a divisor of a?

10

13

18

26

50

here is the explanation provided by MGMAT CAT:
a

b

=

286

100
Now, we must reduce:

a

b

=

143

50



It might be easier to think through the problem if we cross multiply:

50 × a = 143 × b

What does that tell us about a and b? Well, we know that 50, a, 143, and b are all integers. Thus both sides of the equation will be integers (the same integer). For that to be true, both sides of the equation must have IDENTICAL prime factorizations.

We know that the left side of the equation has a 2 and 2 5's in its prime factorization (50 = 5×5×2). Therefore, b must have at least a 2, a 5 and another 5 in its prime factorization. So b is divisible by 50. Furthermore, we know that the right side of the equation has an 11 and a 13 in its prime factorization (143 = 11×13). Therefore, a must have at least an 11 and a 13 in its prime factorization. So a is divisible by 11, 13, and 143.

The question asks about a. We know that a must be divisible by 13.

The correct answer is B.

My Q is ----why can't D be the answer? It is divisible by 13 as well. Is it b/c it requires another prime, 2? And we cannot say for sure, whether a is divisible by that prime?

[Brian@VeritasPrep]

Good question, san2009 - both yours, and the question you reference.

One somewhat easy way to just get a feel for this question is to ask yourself which division problem would give you 2.86. The first one to come to mind would be 286/100, right? Well, you can factor that down by dividing both by 2 to 143/50. At that point, they don't have any common factors (13*11 / 5*5*2), so that's the smallest possible pair of integers that would get you to 2.86.

In that calculation, 26 is not a factor of the numerator, a, which is simply 13*11, or 143. Therefore, 26 does not have to be a factor of a. Because the question asks which MUST BE a factor of a, 26 is incorrect...it very well could be, but need not be a factor.


Another clue here - even if you only had 15-20 seconds to look at this one, you can eliminate D and E, because each of those has a factor represented in another answer choice (50 is divisible by 10; 26 is divisible by 13). Because the question will only allow for one correct answer, and the answer MUST BE a factor of a, it couldn't be either 26 or 50 because then there would be two answers. (if the question asked "which is the largest integer that must be a factor", that would be different...)

[kstv]

a/b = 2.86
it is possible to represent the ratio a/b = 286/100
286 = 2*13*11
100 = 2*5*5
the common factor 2 in the numerator and denominator will cancel out.
a/b can be represented as (13*11)/50 which is also = 2.86 , in that case 2 cannot be a divisor of a

so 13 and 11 are divisors of a.

[san2009]

great, thanks.