Data Sufficiency

Geva@MasterGMAT wrote:to be divisible by 21, x needs to be divisible by 3 and 7

From stat. (1), we get that x is not divisible by 14, which means that it's not divisible by 7. Therefore x is NOT divisible by 21 - but that is sufficient, since you CAN answer the question stem with a single, definite answer "no".

From stat. (2), we get that x is not divisible by 15, which means that it's not divisible by 3. Same here: x is NOT divisible by 21 - but that is sufficient, since you CAN answer the question stem with a single, definite answer "no".

Answer is indeed D.

Great kickoff, but I would be a bit more explicit on how we know that statement (1) shows not divisible by 7 and statement (2) shows not divisible by 3. I could rewrite the problem by only changing the remainders to have the answer be C (I will post that re-write at the bottom of this explanation).

Rephrase the Stem: "Is X divisible by 21?"
- A quick rephrase would be - "is X divisible by BOTH 7 and 3?" (the factors of 21).

Statement 1:
We know that X is not divisible by 14, but we don't know which factor(s) of 14 are causing the problem (the 2, the 7 or both). For example, 21 is not divisible by 14, but that is because 21 is missing a factor of 2 (not 7). 16 is not divisible by 14 either, but this time a missing 7 is the problem. Finally, 19 is not divisible by 14 because BOTH a factor of 2 and of 7 are missing. So let's find a way to see which factors of 14 are causing the issue:

Using the Remainder theorem, we can express X as the following:

X = 14k + 4; where k is any positive integer.

X = 2(7k+2) (further reducing)

Therefore, we know that X has the factor of 2 (if is a multiple of 2) but it must be missing the factor of 7 - therefore X cannot be divisible by 21 (because a 7 and a 3 must be present to be divisible by 21).

Now let's try with the second statement:

Statement 2:
We know that X is not divisible by 15, but we don't know which factor(s) of 15 are causing the problem (the 3, the 5 or both). For example, 20 is not divisible by 15, but that is because 20 is missing a factor of 3. However, 18 is not divisible by 15 either, but this time it is because the 18 is missing a factor of 5. Finally, 23 is not divisible by 15 because BOTH the factors 3 and the 5 are missing. So let's find a way to see which factors of 15 are causing the issue with X:

Using the Remainder theorem, we can express X as the following:

X = 15j + 5; where j is any positive integer.

X = 5(3k+1) (further reducing)

Therefore, we know that X has the factor of 5 (if is a multiple of 5) but it must be missing the factor of 3 - therefore X cannot be divisible by 21 (because a 3 and a 7 must be present to be divisible by 21).

The correct answer is D.


Hope this helps!

Whit



**** EXTENSION PROBLEM ****
So I said I would provide an almost identical example that gives a different answer.

Is the positive integer X divisible by 21?
(1) When X is divided by 14, the remainder is 7
(2) When X is divided by 15, the remainder is 9

From statement (1) we know again that X is not divisible by 14 but we cannot just say that means it is not divisible by 7. The same goes for statement (2), not being divisible by 15 does not necessarily mean not divisible by 3. Let's hit both statements with the remainder theorem equations:

(1) X = 14k + 7
X = 7(2k+1)
- X is actually divisible by 7 this time, but we do not know whether the (2k+1) factor contains a 3, so NOT Sufficient.

(2) X = 15j + 9
X = 3(5j + 3)
- X is actually divisible by 3 this time, but we do not know whether the (5j+3) factor contains a 7, so NOT sufficient.

(1+2) We know that X is a multiple of BOTH 3 and 7, so SUFFICIENT - C.