N = 2^j3^k, where j and k are positive integers. What is N?

1st condition: 2 is a divisor of N, but 4 is not.

This statement tells us that \(j<2\) because, \(2^2 = 4\) and 4 is not a factor of N. Since 2 is a factor of N, it follows that \(j\leq{1}\). Combining the first and second part, and noting that j is a positive number, we get that \(j = 1\). However, since we no have information about k, st. 1 is not sufficient.

2nd condition: N is a divisor of 36, but not of 24.

\(36= 2^2 * 3^2\\
24 = 2^3 * 3^1\)

This statement tell us that k will need to be at least 2 or above in order for N to be a factor of 36. Another way to look at it: If \(k=0\), then j can be any +ve number but we already know that j & k are +ve; thus \(k = 0\) is out. If \(k = 1\), then j can be any value but to be a factor of 36 but not 24; we have to exclude 6 and 12 from possible values of N. Thus, this only leaves those factors which has k with values of 2 or above. But as this st. doesn't give us any information about j, st. 2 is not sufficient.

Combining the two statement, \(j = 1, k\geq{2}\), thus N can be \(2 * 3^2\) = 18, which is also a factor of 36.

Thus, the two statements combined are helpful but neither st. alone is sufficient. (C)