homer wrote:The trick (or catch) is that n itself may not be an integer.

So, if n = (18/5) condition 1 is true - but n/18 = 1/5 => not an integer

The second condition by itself is also not sufficient - as yuo have probably figured out anyways. BUT it tels you that n/6 is an integer. This establishes that _n_ is an integer.

Now - if we go back to condition 1, with the knowledge, that n is an integer, n/18 is an integer is tautologically true.

Hence, C - both conditions are necessary.

rosenjon wrote:I found MGMAT's factor box explanation very helpful for these types of problems. This is a slightly less "math intensive" way to approach it.

Remember that any number is a factor of another number if that number has the same number and type of primes. MGMAT has you put these primes in a "factor box" that represents that number. This makes it very easy to break down the problem and compare factor boxes to see if a number is a factor of another.

So let's look at the question:

Is n/18 an integer? Well, before we even get to the question, let's factor 18.

18 = 9x2 = 3x3x2

So let's rethink the question stem? Is n/(3x3x2) an integer. Or, in other words, if n has 3,3,2 in its "factor box", then n is an integer. If n doesn't have 3,3,2 in its factor box, then n/18 cannot be an integer.

Statement 1 says that 5n/18 is an integer. Well, we know that the prime factorization of 18 = 3x3x2 from above. Since 5n IS AN INTEGER, then 5n must have 3,3,2 in its "factor box". However, since 3,3,2 MUST be a factor of 5n, and 5 is not divisble by 3 or 2, then n must be divisible by 3,3,2.

Our stem statement says that n is divisible by 18 if it has 3,3,2 in its factor box. Statement 1 proves that it does, so statement 1 is sufficient.

Statement 2 asks if 3n/18 is an integer.

The explanation for statement 2 is along the same lines as above.

3n/18 IS an integer.

Therefore, 3n/(3x3x2) must be an integer. If you reduce this, you get n is divisible by 3x2. Therefore, according to this statement, n has 3,2 in its factor box.

However, before we began the problem, we decided that n must have 3,3,2 in its factor box to be divisible by 18. Statement 2 only tells us that it has 3,2.

Therefore, n MIGHT be divisible by 18. We don't know the value of n, so we can't determine whether it is or not. If we knew that n ONLY had 3,2 in its factor box, and nothing else, then the answer would be no, and 2 would be sufficient.

But n has some of the factors of 18 in its factor box, plus some other set of unknown factors. Therefore, n could be divisible by 18, or it might not be. Without knowing the specific value of n, it is impossible to determine.

Therefore, Statement 2 is not sufficient.

The answer is therefore A.