Positive integer x divisible by 24

didieravoaka wrote:Is positive integer x divisible by 24?

1) √x is divisible by 4.

2) x² is not divisible by 9.

For a number to be divisible by 24, the number has to be divisible by the combination of all of the prime factors of 24.

The prime factors of 24 are 2, 2, 2, and 3.

Statement 1 tells us that x is divisible by 4². The prime factors of 4² are 2, 2, 2 and 2. So x has enough 2's in its prime factorization such that it can be divided by 24, but Statement 1 does not include information on whether there are any 3's among the prime factors of x.

Insufficient.

Statement 2 says that x² is not divisible by 9. If x had any 3's among its prime factors, x² would be divisible by 9. So x must not have any 3's among it's prime factors, and therefore there is no way that x could be a multiple of 24.

Sufficient.

The correct answer is B.

didieravoaka wrote:
Is positive integer x divisible by 24?

1) √x is divisible by 4.

2) x² is not divisible by 9.

And there's always good old-fashioned number-picking.

Statement 1: For Case 1, consider x = 16. Not divisible by 24, so the answer to the question is NO. Consider x = 144. Divisible by 24, so the answer is YES. Statement 1 is not sufficient.

Statement 2: For Case 1, consider x^2 = 16, or x = 4. Not divisible by 24, so we have a NO. Consider x^2 = 25, or x = 5. Not divisible by 24, so we have another NO. Now, as Marty noted, if x^2 is not divisible by 9, integer x cannot contain a 3. Without a 3, x cannot be divisible by 24, so the answer will always be NO. Therefore S2 is sufficient on its own. Answer is B.

didieravoaka wrote:Thanks to help.

Plugging in the numbers is the easiest way out to crackdown such problems:

(1) √x is divisible by 4. Here, x = 16 fits but 16 is NOT divisible by 24; and x = 576 also fits but 576 is YES divisible by 24. Hence insufficient!

(2) Since 3 is a factor of 24, therefore had x been divisible by 24, x ^2 must have been divisible by 9, but it’s not. Hence, x is [spoiler]NOT divisible by 24. Sufficient!


Pick B[/spoiler]

One other takeaway here: if a number is divisible by 24, then it's divisible by ALL the factors of 24 (e.g. 1, 2, 3, 4, 6, 8, 12, and 24). So if a number is NOT divisible by a certain factor of 24, it CAN'T be divisible by 24!

The GMAT loves to make you solve these problems with your weak hand: you're using to proving something is divisible, so they challenge you to think of what would make something NOT divisible. It's OK not to think of this the first time, but if you miss a question like this, remember how you were fooled and add a defense to your arsenal.