n is a positive integer less than 300. Is n prime?

Question

(1) There is only one integer between 1 and n/2, exclusive, which divides n completely

(2) The sum of all the factors of n is greater than n + 1.

MofeBhatia · Answer

(1) There is only one integer between 1 and n/2, exclusive, which divides n completely.
Statement 1 implies there is one integer which divides n and that integer is btw 1 and n/2.
This means n is not prime.
Sufficient.

(2) The sum of all the factors of n is greater than n + 1.
Statement 2 implies there is one more factor other than 1 and n itself. For eg- Factors of 5 are 1 and 5 
Sum of factors = 5 +1 = 6 or n+1 , If sum of all factors of n are greater than n+1 , it means n has more factors apart form 1 ,n . N is not prime.
Sufficient.

Answer D.

TarPhi · Answer

There is only one integer between 1 and n/2, exclusive, which divides n completely. 
For a number to be prime, it should only be divisible by 1 and itself.
Statement 1 implies there is an integer which divides n that lies between 1 and n/2.
This means n is not prime.
 Sufficient.

The sum of all the factors of n is greater than n + 1. 
For any prime number, the sum of the factors with be 1+n. For eg. For 5 = 1+5 = 6
If the sum of all factors of n is greater than n+1, it means implies that n has more factors apart from 1 and itself (n). 
N is not prime.
 Sufficient

Answer D.

TestPrepUnlimited · Answer

Statement 1: 
If n has a factor of 2, n/2 would also be a factor of n. Similarly, if n has a factor of 3, n/3 would be a factor of 3. If n > 6, both 3 and n/3 are between 1 and n/2, this cannot be the case (the same applies for any other factor between 1 and n/2 exclusive). The only case viable here is having the factor of 2 and n/2, since n/2 is excluded from the range and 2 is that "one integer between 1 and n/2". So n is not prime, and we can also note  n = A prime number * 2  or n = 8. Sufficient.

Statement 2: 
If n was prime, n would only have factors of n and 1. The sum of factors would be n + 1. Since the sum of factors is greater, n cannot be prime. Sufficient.

Ans: D

ccooley · Answer

This looks like a tough problem on the surface. First, if n was less than some smaller value (like less than 50), I might take a moment and jot down all of the primes that could possibly fit, just so I had them handy. But because n is less than 300, that's still a lot of different primes. We'll have to start with the statements instead.

Neither statement looks really easy to me. However,  Statement 1  ends up being deceptively useful. Since there's an integer between 1 and n/2 that divides n, that means n can't be prime, so the answer to the question is "no." The fact that it's  only  one integer is a red herring. You could easily waste a lot of time trying to find a value of n with this property, but you really don't need to, since you have enough to answer the yes/no question you were asked. This statement is  sufficient .

Statement 2 : the sum of all the factors of n is greater than n+1. Having looked at statement 1, I'm already a bit suspicious that this statement may be trying to waste my time. Let's think about primes in general before we test cases. If n  was  a prime, its only factors would be n and 1. Therefore, the sum of its factors would be n+1 exactly. However, we know this isn't true. Therefore, n must not be prime and the answer to the question must be "no." This statement is also  sufficient .

The correct answer is  D .

DhruvS · Answer

(1) There is only one integer between 1 and n/2, exclusive, which divides n completely.
Statement 1 implies there is one integer which divides n and that integer is btw 1 and n/2.
This means n is not prime.
Sufficient.

(2) The sum of all the factors of n is greater than n + 1.
Statement 2 implies there is one more factor other than 1 and n itself. For eg- Factors of 5 are 1 and 5
Sum of factors = 5 +1 = 6 or n+1 , If sum of all factors of n are greater than n+1 , it means n has more factors apart form 1 ,n . N is not prime.
Sufficient.

Answer D.

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n is a positive integer less than 300. Is n prime?

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GMAT Data Sufficiency (DS) Questions

n is a positive integer less than 300. Is n prime?

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