If integer n is greater than 1, Does n have more than two distinct fac

Question

If integer n is greater than 1, Does n have more than two distinct factors?

(1) 11! + 2 < n < 11! + 11
(2) n is not a prime number.

EMPOWERgmatRichC · Accepted Answer

Hi gmatmania17,

The GMAT will include a certain number of questions that test you on concepts that you already know, but in ways that you're not used to thinking about.

For example, you can probably factor 2X + 4 into 2(X+2). Since both terms (the "2X" and the "4") are even, your eye "catches" that pattern and you can rewrite the information. That SAME idea is in this DS question, but you might not notice it...

We're told that N is an INTEGER > 1. We're asked if N has MORE than 2 distinct factors. This is a YES/NO question.

Before we get to the two Facts, here is how you should be thinking about the question itself:

If N = 2, 3, 5, 7, 11....then it has JUST 2 factors and the answer to the question is NO
If N = 4, 6, 8, 9, 10....then it has MORE than 2 factors and the answer to the question is YES

This question is essentially asking if N is a PRIME number or not.

Fact 1: 11! + 2 < N < 11! + 11

11! might seem like a "scary" number, but we're not required to calculate it. To properly deal with Fact 1, we must FACTOR it though. Here's how:

11! + 3 = (11)(10)(9)(8)....(3)(2)(1) + 3

Here, a "3" appears in both terms, so we can factor it out...

3[(11)(10)(9)(8)....(4)(2)(1) + 1]

This big number is NOT prime, since it's divisible by 1, 3, a really big number I won't calculate, and itself.

You'll find that 11! + 4 follows this same pattern...
So does 11! + 5
So does 11! + 6
Etc....all the way up to
11! + 10

So ALL of these numbers have MORE than 2 factors and the answer to the question is ALWAYS YES.
Fact 1 is SUFFICIENT.

Fact 2: N is NOT a prime number

Since we were already told that N > 1 and an integer, the fact that it's NOT prime gives us everything we need to answer the question logically. If you want to write down a few quick examples though, you can...

N could be 4, 6, 8, 9, 10, 12

ALL of these numbers have MORE than 2 factors, so the answer to the question is ALWAYS YES.
Fact 2 is SUFFICIENT.

Final Answer:

Show Spoiler

D

GMAT assassins aren't born, they're made,
Rich

littlewarthog · Answer

The question can be rephrased to: Is x prime? Now let's look at the statements:

Statement 1: 11! has all integers <=11 as its factors. Therefore, adding an integer between 2 and 11 to 11! will always share at least the integer itself as a common factor with 11! and therefore n cannot be a prime. Note that only for 11!+1, n could be prime, as consecutive integers are co-prime, but this case is excluded by the statement. Sufficient.

Statement 2: The statement tells us that n is not a prime, therefore it is sufficient.

Answer D.

Bunuel · Answer

Similar questions to practice: if-x-is-an-integer-does-x-have-a-factor-n-such-that-100670.html does-the-integer-k-have-a-factor-p-such-that-1-p-k-126735.html for-any-integer-n-greater-than-1-n-denotes-the-product-of-168575.html does-integer-n-have-2-factors-x-y-such-that-1-x-y-n-165983.html if-z-is-an-integer-is-z-prime-128732.html does-p-have-a-factor-x-where-1-x-p-and-x-and-p-are-positive-integers-188773.html Hope this helps.

MathRevolution · Answer

Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution

if integer n is greater than 1, Does n have more than two distinct factors?

(1) 11! +2<n<11!+11
(2) n is not a prime number. 
 
There 1 variable in the original condition. In order to match the number of variables and the number of questions, we need 1 equation. Since the condition 1) and 2) each has 1 equation, there is high chance D is the answer. 
Using both the condition 1) and 2):

In case of the condition 1), we get n=11!+3, 11!+4, 11!+5,....,11!+10. Then, n=3 , 4 , .... and in any case, n is not a prime number. So, the number of distinct factors is at least (1+1)(1+1)=4. Therefore, the answer is ‘yes’ and the condition is sufficient. The correct answer, hence, is D.

For cases where we need 1 more equation, such as original conditions with “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 59 % chance that D is the answer, while A or B has 38% chance and C or E has 3% chance. Since D is most likely to be the answer using 1) and 2) separately according to DS definition. Obviously there may be cases where the answer is A, B, C or E.

pepo · Answer

If integer n is greater than 1, does n have more than two distinct factors?

1) 11! + 2<n<11! + 11

2) n is not a prime number

Divyadisha · Answer

n is a positive integer >1.

Question asked- n is a prime number or not?

1) 11! + 2<n<11! + 11

if n is 11!+3, then it surely has three factors- 1, 3 and 11!+3

Similarly all the numbers between 11! + 2<n<11! + 11 will surely have three factors. And hence will not be prime.

Sufficient

2) n is not a prime number. Straight sufficient.

D is the answer

Bunuel · Answer

Merging topics. Please refer to the discussion above.

mdacosta · Answer

I loved your explanation, one question: ignoring the inequality, when would a number be prime? Meaning 11! + 2 to 11! + 10 are not primes, but if i keep adding +1 to the integer when would I get a prime #? My thought is that even 11! + 13 would not be prime if i'm understanding your explanation b/c it would be divisible by "a big number" and 13, is that correct? If so, then would you ever get a prime number?

Thanks!

EMPOWERgmatRichC · Answer

Hi mdacosta,

You ask an interesting question, but it's not one that the GMAT would ask of you under these circumstances. The Quant section of the GMAT is not a 'math test'; it's a critical thinking test that requires you know some math rules and do some basic calculations. As such, you won't be expected to do lots of complex math to answer any Quant questions on Test Day (although you might choose to take a 'math heavy' approach, there is almost always an easier, Tactical way to get to the correct answer).

In this question, there was an easy way to tell that (11! + 3) was not prime (without doing lots of calculations): the rules of factoring. In your example, there's isn't an obvious way to tell whether 11! + 13 is prime or not (beyond doing lots of math with a calculator), so the GMAT wouldn't ask you that question.

GMAT assassins aren't born, they're made,
Rich

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