apoorva.srivastva wrote:Is the positive integer n divisible by 18?
1.) n^2 is divisible by 18
2.) 2n is divisible by 18
From the stem, we know that n is a positive integer. Our task: to determine if n is divisible by 18 or, in other words, if n has, among its prime factors, 2, 3 and 3.
1) n^2 is divisible by 18.
So n^2 has factors of 2, 3 and 3. Does this mean that n also has those factors? No: we know that n^2 will have pairs of primes, so n^2 has to include at least one more "2", but that doesn't mean that n has to have two "3"s.
For example, n^2 could be 2*2*3*3 = 36, which means that n=6, which isn't a multiple of 18.
Or, n^2 could be 2*2*2*2*3*3*3*3 = 36^2, which means that n=36, which is a multiple of 18.
Insufficient, eliminate A and D.
2) 2n is divisible by 18.
Well, if 2n is divisible by 18, then n must be divisible by 18/2 = 9.
We could pick n=9 to get a "no" answer or n=18 to get a "yes" answer.
Insufficient, eliminate B.
Since neither statement was sufficient alone, we need to combine.
From (1), we know that n must be even.
From (2), we know that n is a multiple of 9.
So, together, we know that n is an even multiple of 9.
Is every even multiple of 9 divisible by 18? YES - sufficient, choose C.
