Data Sufficiency

[email protected] wrote:In a sequence of 13 consecutive integers, all of which are less than 100, there are exactly three multiples of 6. How many integers in the sequence are prime?

(1) Both of the multiples of 5 also in the sequence are multiples of either 2 or 3.
(2) Only one of the two multiples of 7 also in the sequence is not a multiple of 2 or 3

While we can solve with trial and error method; is there any method to do this?

Dear shibsriz,
I'm happy to respond.

Yes, this one is a trial & error question to some extent, although one can use number properties as shortcuts. FWIW, here's a blog on DS tips in general.
https://magoosh.com/gmat/2013/gmat-data- ... ency-tips/

The fact that this set of 13 consecutive integers contains THREE multiples of 6 is a huge hint. This means the string of consecutive integers must start and end on a multiple of six.

My general strategy would be to try the lowest and highest strings of 13 consecutive integers possible, given the constraints of each statement, only because prime numbers start out more dense among lower numbers and more spread out as the numbers get bigger.

Here's my thought process:

Statement #1: Both of the multiples of 5 also in the sequence are multiples of either 2 or 3.
Obviously, the multiple of 10 in the set will be a multiple of 5 that is a multiple of 2. This means, the odd multiple of 5, which is not a multiple of 10, must be a multiple of 3 --- that gives us only 15, 45, and 75 as choices.
Set A = {6 - 18} --- this set obeys this statement, and has four primes (7, 11, 13, 17)
Set B = {72 - 84} --- this set obeys this statement, and has three primes (73, 79, 83)
Different sets consistent with the statement give different answers to the prompt. This statement, alone and by itself, is not sufficient.

Statement #2: Only one of the two multiples of 7 also in the sequence is not a multiple of 2 or 3
Fortunately, Sets A & B both satisfy this condition as well. Different sets consistent with the statement give different answers to the prompt. This statement, alone and by itself, is also not sufficient.

Combined:
Sets A & B both satisfy both statements. Different sets consistent with both statements give different answers to the prompt. Both statements together are not sufficient.

Answer = [spoiler](E)[/spoiler]

Notice: if we can pick examples for Statement #1 that also work for Statement #2, that's a huge time-saver.

Does all this make sense?
Mike