is n odd?

[mkhanna]

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

[mp2437]

E. Use examples:

Statement 1 isn't sufficient by itself since it could be 3,6,or 9, examples of odd and even numbers, so you're not sure which one it is.

Statement 2 isn't sufficient by itself since it could be 3 or 7.

For N = 3: (2*3 = 6 has factors of 1,2,3,6 which is a total of 4 factors, as opposed to N,which has a total of 2 factors (1 and 3)
For N = 7: (2*7 = 14 has factors of 1,2,7,14 as opposed to N=7, which has factors of 1 and 7).
For N = 9: (2*9 = 18 has factors of 1,2,3,6,9,18, and N=9 has a total of 3 factors - 1,3,9)

Using information from both statements, you could determine that neither are sufficient on their own, and if you use both, you still end up with no definite answer (could be N = 3 or N = 9), so answer is choice E.

[zuleron]

actually B is the answer. They are asking whether N is odd, not what the value of N is. Pick a few numbers and see... 2 * an odd number always has 2 times as many factors as the original odd number. This is not true for even numbers. So B is sufficient.

[mp2437]

sorry, misread question!!! In that case, answer is B as per zuleron's comments.

[zuleron]

The more interesting question is why do odd numbers behave this way and even numbers do not...

[Ian Stewart]

The more interesting question is why do odd numbers behave this way and even numbers do not...

I posted an explanation to gmatclub recently:

gmatclub.com/forum/odd-integer-72930.html

I'll quote my post below, but it refers back to an earlier post in the gmatclub thread:

There must be a general rule behind this to avoid plugging numbers. Anyone know?

There is a general rule here, which we can arrive at by extending the logic in maliyeci's excellent explanation above. I could explain this abstractly, but it's probably easier to take a specific example - let's use the number 72 = (2^3)(3^2). Now, this number has 12 factors in total, three of which are odd:

1, 3, 3^2

Now, if we multiply each of the numbers above by 2^1, we get three even divisors of 72, and the same will happen if we multiply these numbers by 2^2 or 2^3. So 72 has three odd divisors, and nine even divisors:

1, 3, 3^2
2, 2*3, 2*3^2
2^2, (2^2)*3, (2^2)(3^2)
2^3, (2^3)*3, (2^3)(3^2)

Notice that we have three times as many even divisors as odd divisors because the power on the 2 in the prime factorization of 72 is 3; that guarantees that we have three even divisors for every odd divisor. You could use this logic for any number, of course, from which we have the following general rule:

* The ratio of the number of even divisors of x to the number of odd divisors of x is always equal to the power on the 2 in the prime factorization of x.

So, if the power on the 2 in the prime factorization of x is equal to 1, we have an equal number of odd and even divisors. If the power is greater than 1, we have more even divisors than odd divisors.

[zuleron]

Thanks Ian! I learn something new everyday!

[farooq]

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

I know it looks simple, but I am unable to understand the a mathematical meaning of second statement.

I took it as 2N is divisible by twice as many positive integers as N. It means 2N/ (no. of integers that N has)

So for example,
N = 3, expression becomes 2N/3 = 2*3/3.
N = 6, expression becomes 2N/6 = 2*6/6.

As per above understanding I marked E