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Could anyone help me with this one?
Q19:
Is the integer n odd?
(1) n is divisible by 3.
(2) 2n is divisible by twice as many positive integers as n.
A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is
sufficient.
D. EACH statement ALONE is sufficient.
E. Statements (1) and (2) TOGETHER are NOT sufficient.
Answer

1) INSUFF
Integer n could be 3 or 6 for example
2) SUFFICIENT
The square of any odd prime number has exactly 3 factors. If you double the square of a prime number, the number of factors also doubles. Try n = 9 confirm.

1) INSUFF Integer n could be 3 or 6 for example 2) SUFFICIENT The square of any odd prime number has exactly 3 factors. If you double the square of a prime number, the number of factors also doubles. Try n = 9 confirm.

B. OA?

I think it's E; While your observation about square of odd primes having exactly 3 factors (and the double of the square having 6 factors) is true the essential point is how can u generalize if the odd prime being squared itself is a multiple of 3 (as required in the que)

For n = 9 (square of 3) you are right and 3 is div by 3.

Take n = 5 which is NOT div by 3; the same obs applies; 5^2 = 25; three divisors for 25 - 1, 5, 25 and 6 divisors for 25*2 = 50; 1,2,5,10, 25,50

Is the integer n odd?
(1) n is divisible by 3.
(2) 2n is divisible by twice as many positive integers as n.

(1) n is divisible by 3
------------------------
n has 3 as a factor. So what ? What determins if n is odd is whether n has 2 as a factor or not.

INSUFFICIENT

(2) 2n is divisible by twice as many positive integers as n
------------------------------------------------------------------
How does this help ? let's try picking numbers
odd integer 9: divisible by 1,3,9. For 2x9 = 18: 1,2,3,6,9,18
Here 2n is divisible by factors of n and each of 2p where p is a factor of n.

[quote="yuefei"]1) INSUFF
Integer n could be 3 or 6 for example
2) SUFFICIENT
The square of any odd prime number has exactly 3 factors. If you double the square of a prime number, the number of factors also doubles. Try n = 9 confirm.

what does this rule have anything to do here? I dont get the point

If n is odd: if n is odd, then all factors of n are odd. Thus, 2n is divisible by all factors of n plus each of 2p where p is each factor of n.

If n is even: 2n will not lead to twice as many factors as n since n is already even and has 2 as a factor and probably another even factor.

in other words,
if n is odd and has factors : 1, k, w, and n, all of which are odd. Then 2n will have factors 1, 2 , k, w, 2k, 2w, and 2n. Twice as many factors is because 2 for the 1 factor, 2n for the n factor and 2 times each other factor.

The square of any prime number has exactly 3 factors; 2n has 6 factors. This was the first rule that came to mind - since 2 is the only even prime, this allowed me to quickly answer the question on whether n is an odd number.

Could anyone help me with this one? Q19: Is the integer n odd? (1) n is divisible by 3. (2) 2n is divisible by twice as many positive integers as n. A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. D. EACH statement ALONE is sufficient. E. Statements (1) and (2) TOGETHER are NOT sufficient. Answer

I think it's B.... Info 1 doesnt give unique answer while basis info 2 we can deduce that n has to be a prime number so - n has to be ODD