MGMAT CAT Question on Num Prop

[sudaif]

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?

[akhp77]

Ratio of a to b
a / b = 143 / 50

Minimum value of a could be 143 = 11*13
OR
a = 11*13*1, 11*13*2, 11*13*3, 11*13*4

In question, it is mentioned that "must be divisible"

So, in any case a is divisible by 13 and 11.

a could be divisible by 26 but not always. So, Eliminated

[tim]

akhp is right. It may help you to list the possible values of a. a can be 143, 286, 429, 572, etc. Not ALL of these are divisible by 26, even though some are. When the question asks what a MUST be divisible by, that means we need a number that ALWAYS goes into a no matter which value of a we choose..