Critical Reasoning

Hi All,

Could anyone please explain why statement 1 assumes b to be prime..?
Greatest factor of x less than itself can be non-prime also.?

Thanks,
Mallika

Hey Mallika,

This is in the wrong forum. You'll want to post Data Sufficiency questions here: https://www.beatthegmat.com/data-sufficiency-f7.html

To answer your question, let's first look at the construction:

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?

I find this problem overly wordy and a bit too intentionally confusing than most GMAT DS, so I definitely wouldn't fault you for some confusion! To break this thing down, let's examine the givens and the simple question:

GIVENS:

-b and x are positive integers
-b is the greatest divisor of x
-b < x
-divisors of x = between 1 and x

QUESTION:

Is the SUM of the divisors of x (between 1 and x) LARGER than 2b?

This is a "YES/NO" question, so a statement is sufficient if it allows us to answer the question 100% YES or 100% NO.

As written, we definitely need more information about x and b. What are the possible values for these two variables? A sufficient statement should limit the values of x and b such that it allows the question to only be answered one way. Ask yourself: does this statement limit x and b ENOUGH?

Let's tackle the statements!

(1) b^2 = x

Picking number is a good strategy here, since we do not know what x and b are. Let's say b = 2 and x = 4. The only divisors of 4 between 1 and 4 is 2, so the sum is 2. In this case, the sum is NOT larger than 2b, or 2(2) = 4. The answer to the question is NO.

Let's choose a larger number for b and see if we get a different result. Let's say b = 4, and x = 16. Already this option doesn't work, because it contradicts the givens. 4 is NOT the largest divisor of 16 that is less than 16, so this isn't a possible set. REMEMBER THAT YOUR PICKED NUMBERS MUST ABIDE BY THE GIVENS!

What about b = 3 and x = 9. That would work, since 3 is the largest divisor of 9 that is less than 9. We get the same result as when we chose b = 2 and x = 4. The sum of the divisors of 9 between 1 and 9 is 3. 3 is NOT less than 2b (2*3) = 6. The answer is always NO.

It's not the "PRIME"-ness that matters in this question, it's that it just so happens that the numbers that work with the givens are prime: 2, 3, etc.

VivianKerr wrote:Hey Mallika,

This is in the wrong forum. You'll want to post Data Sufficiency questions here: https://www.beatthegmat.com/data-sufficiency-f7.html

To answer your question, let's first look at the construction:

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?

I find this problem overly wordy and a bit too intentionally confusing than most GMAT DS, so I definitely wouldn't fault you for some confusion! To break this thing down, let's examine the givens and the simple question:

GIVENS:

-b and x are positive integers
-b is the greatest divisor of x
-b < x
-divisors of x = between 1 and x

QUESTION:

Is the SUM of the divisors of x (between 1 and x) LARGER than 2b?

This is a "YES/NO" question, so a statement is sufficient if it allows us to answer the question 100% YES or 100% NO.

As written, we definitely need more information about x and b. What are the possible values for these two variables? A sufficient statement should limit the values of x and b such that it allows the question to only be answered one way. Ask yourself: does this statement limit x and b ENOUGH?

Let's tackle the statements!

(1) b^2 = x

Picking number is a good strategy here, since we do not know what x and b are. Let's say b = 2 and x = 4. The only divisors of 4 between 1 and 4 is 2, so the sum is 2. In this case, the sum is NOT larger than 2b, or 2(2) = 4. The answer to the question is NO.

Let's choose a larger number for b and see if we get a different result. Let's say b = 4, and x = 16. Already this option doesn't work, because it contradicts the givens. 4 is NOT the largest divisor of 16 that is less than 16, so this isn't a possible set. REMEMBER THAT YOUR PICKED NUMBERS MUST ABIDE BY THE GIVENS!

What about b = 3 and x = 9. That would work, since 3 is the largest divisor of 9 that is less than 9. We get the same result as when we chose b = 2 and x = 4. The sum of the divisors of 9 between 1 and 9 is 3. 3 is NOT less than 2b (2*3) = 6. The answer is always NO.

It's not the "PRIME"-ness that matters in this question, it's that it just so happens that the numbers that work with the givens are prime: 2, 3, etc.

Many thanks for the explanation Vivian!! I now get that out of the numbers we pick, the ones that satisfy the statement happen to be prime!

Thanks again,
Mallika

You're very welcome, and "thanks" for the thanks!