Is the integer n a multiple of 15 ? (1) n is a multiple of 20 (2) n

Start with statement 1. It is not sufficient because for a number to be a multiple for 15, it should be divisible by both 3 and 5. Clearly, 20 does not have 3 as a factor. Hence, statement 1 is not sufficient.

Statement 2 says that n+6 is a multiple of three. If you break it down, it is basically saying that n is a multiple of 3. But that we don't know whether n is a multiple of 5 or not. Hence, this is not sufficient.

If you combine both 1 and 2, you can conclude that n has both 5 (from statement 1)and 3 (from statement 2) as factors. Hence, C is the answer.

Q: Is n a multiple of 15?

This can be solved by prime factors.

Question can be rephrased as; are at least both 3 and 5 prime factors of n

3 and 5 are factors of 15. If n also contains at least both 3 and 5 as factors, it must be divided by 15.

1. n is a multiple of 20.
Prime factors of 20 are 2*2*5

This tells us that n is definitely a multiple of 5. But, there is no 3 among its factors. We must have at least both 3 and 5 as factors for n to be a definite multiple of 15.

We also can't definitely tell that n is not a multiple of 15.

e.g.
n=40- Multiple of 20. NOT a Multiple of 15.
n=60- Multiple of 20. Also a multiple of 15.

So, the prime factors 2*2*5 tell us that n is NOT definitely a multiple of 15. It may or may not be a multiple of 15. NOT SUFFICIENT.

2. (n+6) is a multiple of 3.

6 is a muliple of 3, so n must also be a multiple of 3.

if (a+b) is a muliple of x and b is a multiple of x, then "a" must be a multiple of x
if (a-b) is a muliple of x and b is a multiple of x, then "a" must be a multiple of x
conversely also true,
if "a" is a multiple of x and "b" is a multiple of x,
then (a+b) must be a multiple of x
also; (a-b) must be a mutiple of x

So, we know n is definitely a multiple of 3. i.e. 3 is a factor of n. But, n is not necessarily a multiple of 15.

For "n" to be a multiple of 15, it must have at least both 3 and 5 as factors.

e.g.

6- multiple of 3. Not a multiple of 15.
30- multiple of 3. Also a multiple of 15.
NOT SUFFICIENT.

Using both statements;
We know 5 and 3 are both factors of n. Thus, n must be a multiple of 15.

SUFFICIENT.

Ans: C

Is the integer n a mulltiple of 15 ?

(1) n is a multiple of 20 --> now, if \(n=0\) then the answer will be YES but if \(n=20\) then the answer will be NO. Not sufficient.

But from this statement we can derive that as \(n\) is a multiple of 20 then it's a multiple of 5.

2) n + 6 is a multiple of 3 --> again, if \(n=0\) then the answer will be YES but if \(n=3\) then the answer will be NO. Not sufficient.

But from this statement we can derive that \(n\) is a multiple of 3 (\(n+6=3q\) --> \(n=3(q-2)\), for some integer \(q\):).

(1)+(2) \(n\) is a multiple of both 5 and 3 thus it must be a multiple of 3*5=15. Sufficient.

Answer: C.
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Thanks a lot guys. Really put it into perspective. I was a little confused with how to use the second statement but how fluke broke it down makes sense. I remember the property that mgmat gave. When a is a multiple of x and b is a mutiple of x. a + b will be a multiple of x.

For n to be a multiple of 15, n must be a multiple of 3 and also a multiple of 5

Statement 1: n is a multiple of 20
This statement confirm that n is a multiple of 5 but doesn't confirm whether n is a multiple of 3 or not hence
NOT SUFFICIENT

Statement 2: n+6 is a multiple of 3
This statement confirm that n is a multiple of 3 but doesn't confirm whether n is a multiple of 5 or not hence
NOT SUFFICIENT

Combining the two statements
n is a multiple of 3 and 5 both hence
SUFFICIENT

Answer: Option C

If n=40

Statement 1: true, because 40 is a multiple of 20
Statement 2: true, because 40 + 6 = 46, which is a multiple of 3
However n is not a multiple of 15.

If n=60

Statement 1: true, because 60 is a multiple of 20
Statement 2: true, because 60 + 6 = 66, which is a multiple of 3
In this case n is a multiple of 15.

So, I fail to understand why answer C is correct, when from the example above we can observe that the statements put together can yield different results for N which are and are not multiples of 15.
In my view E should be the correct choice.
Can someone please explain what is wrong in my reasoning?
Thanks in advance!

46 is not a multiple of 3.
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I was bit confused about the sign in statement 2. It can pass as division as well.
Just wondering can we face such confusing signs on the GMAT

My understaning is that when you combine (1) and (2) and do a LCM of it LCM(20,3) you will get 2*2*5 and 3 --> 2*2*5*3 = 60.

N will multiple of 15 since it is multiple of 60 or you can see that 60 prime factorization contains 3 and 5.

Question Stem Analysis:

We need to determine whether n is a multiple of 15. Notice that if n is a multiple of 3 and 5, then n is also a multiple of LCM(3, 5) = 15.

Statement One Alone:

\(\Rightarrow\) n is a multiple of 20

Since n is a multiple of 20, n is a multiple of 5. However, without knowing whether n is also a multiple of 3, we cannot determine whether n is a multiple of 15. For instance, if n = 20, n is not a multiple of 15. If n = 60, n is a multiple of 15. Since there are more than one possible answers, statement one alone is not sufficient.

Eliminate answer choices A and D.

Statement Two Alone:

\(\Rightarrow\) n + 6 is a multiple of 3

Since n + 6 is a multiple of 3, we can write n + 6 = 3k for some integer k. Then, we can write n = 3k - 6 = 3(k - 2). We see that n is a multiple of 3. However, without knowing whether n is also a multiple of 5, we cannot determine whether n is a multiple of 15. For instance, if n = 3, n is not a multiple of 15. If n = 15, n is a multiple of 15. Since there are more than one possible answers, statement two alone is not sufficient.

Eliminate answer choice B.

Statement One and Two Together:

From statement one, we know n is a multiple of 5. From statement two, we know n is a multiple of 3. Since n is a multiple of both 3 and 5, n is a multiple of 15. Statements one and two together are sufficient.

Answer: C
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