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

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