Primes and Divisibility 'Is this an Integer' Question

[hsiegelbaum]

Know you can not post commentary on how to solve official guide questions, so below is my original adaptation of a format of question seen in the official guide, to the point that it is an original problem, (variables have been changed but I believe will still work and where as before it was a word problem, the example below is all algebra). Have been unable to figure out how to do efficiently (without plugging in and trying scenarios):

Data Sufficiency
Given 5<a<11<b is b/a an integer?
1) 5b/a = integer
2) 11b/a = integer

The question this might echo is 12th Ed OG DS +128

Thanks.

[hsiegelbaum]

Please confirm this is correct:

1) 5b/a = int
- Open ended as it could be the 5 helping the item be divisible w some help from b (IE if m was 5*2 and there was a 2 in b's factor tree) or help could come only from b (IE if a was 7 and b was some multiple of 7) so inconclusive

2) 11b/a = int
- a is a factor of 11b but we know that a is less than 11 which is prime, so a must be a factor of the other part of 11b, which means a is a factor of b

so answer is B

[hsiegelbaum]

in last post
(IE if m was 5*2 and there was a 2 in b's factor tree)

should be:
(IE if a was 5*2 and there was a 2 in b's factor tree)

[Ben Ku]

hsiegelbaum has some good thoughts.

For (1), if a is a multiple of 5, for example 10, then b/a may or may not be an integer. (e.g. b = 15, a = 10 results in non-integer, but b = 14, a = 10 results in integer). So it's insufficient.

For (2), since no value of a cancels with 11, then if 11 (b/a) = integer, then b/a must be an integer. So it's sufficient, and (B) is correct.