Data Sufficiency

Let b and x be positive integers. If b is the greatest divisor of x that is less than x, is the sum of the divisors of x, which are less than x itself and greater than one, greater than 2b?

(1) b^2 = x
(2) 2b = x

A)Statement (1) ALONE is sufficient, but statement (2) is not sufficient.
B)Statement (2) ALONE is sufficient, but statement (1) is not sufficient.
C)BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D)EACH statement ALONE is sufficient.
E)Statements (1) and (2) TOGETHER are NOT sufficient.

Does the question anywhere seem to tell that x is the perfect square of a prime number ??

for 1)

If we take x = 25 and b = 5 then sum of divisors of x other than 1 and x is 5, which is not > 2b. So NO.
If we take x = 36 and b = 6 then the sum will be 2+3+4+6+9+12+18 > 2b . SO YES

INSUFFICIENT.

for 2)

If x = 4 , b = 2 then sum of divisors = 2, which is not > 4. SO NO.
If x = 18 , b = 9 then sum of divisors = 2+3+6+9= 20 > 18. SO YES

INSUFFICIENT.

Taking both together,

b^2 = 2b => b = 0 or b = 2. given b is positive so b = 2 and x = 4 and so sum of divisors of 4 = 2, which is not > 4.
so NO.

SUFFICIENT.

The solution says x is perfect squesre of prime number so in (1) we can take x as 4,9,25 all of which give same answer and hence sufficient.
I dont understand how it is only prime.
Please explain.

prachi18oct wrote:Let b and x be positive integers. If b is the greatest divisor of x that is less than x, is the sum of the divisors of x, which are less than x itself and greater than one, greater than 2b?

(1) b^2 = x
(2) 2b = x

A)Statement (1) ALONE is sufficient, but statement (2) is not sufficient.
B)Statement (2) ALONE is sufficient, but statement (1) is not sufficient.
C)BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D)EACH statement ALONE is sufficient.
E)Statements (1) and (2) TOGETHER are NOT sufficient.

Does the question anywhere seem to tell that x is the perfect square of a prime number ??

The solution says x is perfect squesre of prime number so in (1) we can take x as 4,9,25 all of which give same answer and hence sufficient.
I dont understand how it is only prime.
Please explain.

The question says that b is positive and is the greatest factor of x that is less than x.

Ok.

Now Statement 1 says that b² = x.

With the information given in the question combined with the information given in Statement 1, we can determine that b must be a prime number.

Why?

Because if b were not prime, and b² = x, then there would have to be some factor of x which is greater than b.

Here are some examples.

3² = 9 So if b = 3 and x = 9, then b is the greatest factor of x and b² = x.

Now let's try it with a non prime b.

6² = 36. The problem is 6 can be factored into 3 x 2. So we can rearrange those factors to get 2 x 18 = 36, 3 x 12 = 36. and 4 x 9 = 36.

If b² = x and b is non prime, there will always be a way to rearrange the prime factors of b to get a factor of x that is greater than b.

So the only way the constraints in the question and the constraint in Statement 1 can all be correct is if b is prime and x is the square of a prime number.

Hmm, what a cool, tricky question.

After reading your reply, I realised that I completely misunderstood the question.
The question stem says "b is the greatest divisor of x that is less than x".

I missed the most important part "greatest divisor".
Just to alleviate my curiousity, are both the statements together also sufficient ?

If b² = 2b = x, and b is positive, then b can have only one value, which is 2, and x can only have one value, which is 4.

So yes, as you figured out, even without the greatest divisor constraint, the two statements together are sufficient.

such a complex question statement to understand.

One thing worth remembering here: b is a factor of x. Let's consider a couple cases.

If x is prime, then b = 1.
If x is NOT prime, then x = b * (the smallest factor of x that's greater than 1).

For instance, if x = 6, then the factors are 1, 2, 3, 6, and x = 3 * 2. If x = 45, then b = 15 and x = 15*3, etc.

Let's call c the smallest factor of x that is greater than 1. Therefore if x is NOT prime, we have x = bc.

Now let's take S1, which gives us b² = x.

If x is NOT prime, then bc = x, so bc = b² and b = c. This means that the GREATEST and LEAST factors of x (that are not 1 or x itself) are equal ... hence x only has ONE factor between 1 and itself. That means x is the square of a prime.

If x IS prime, then b² = prime and b = √prime. But this can't be an integer, so x can't be prime.

So we learn that x MUST be the square of a prime, and b must be prime. Once we learn this, we learn that the sum of the factors of x = 1 + b + x, so the sum of the factors (not including 1 and x) = b. b is never greater than 2b, so S1 is sufficient.