Data Sufficiency

For any positive integer x, the 2-height of x is defined to be the greatest non-negative integer n such that 2^n is a factor of x. If k and m are positive integers, is the 2-height of k greater than the 2-height of m?

1. k>m
2. k/m is an even integer



Answer is B


Thanks

Stmt I

k=6 m=4

height of k is 2^1
height of m is 2^2

NO

k=8 m=4

height of k is 2^3
height of m is 2^2

YES

INSUFF


Stmt II

k/m is an even integer (k has to be greater than m and even)

m can be odd or even. In either case the height of k is greater than height of m

SUFF

B)

IMO B

got it, i misunderstood the question

thanks!

Why aren't we considering odd numbers here.

Why aren't we considering odd numbers here.

We can. The height of any odd number would be 2^0 though. In my statement II I had explained. k/m is an even integer (k has to be greater than m and even) m can be odd or even. In either case the height of k is greater than height of m

Let me know if u still hv questions

The 2-height of a positive integer is simply the number of 2's in the integer's prime factorization. For example, the prime factorization of 12 is 2*2*3 (two 2's), and so the 2-height of 12 is 2. The prime factorization of 32 is 2*2*2*2*2 (five 2's), and so the 2-height of 32 is 5. The prime factorization of 33 is 3*11 (zero 2's), and so the 2-height of 33 is 0.

Statement 1.
That k > m does not necessarily mean that k has more 2's in its prime factorization than m has. For example, if k = 12 and m = 10, the answer to the question is yes. But if k = 14 and m = 12, the answer to the question is no. Insufficient.

Statement 2.
If k/m is even, then k must have at least one more 2 in its prime factorization than m has. For example, if k = 12 and m = 2 (k/m = 12/2 = 6, which is even) then the answer to the question is yes. And if k = 10 and m = 5 (k/m = 10/5 = 2, which is even), then the answer to the question is yes. The answer to the question will always be yes. Sufficient.

The correct response is B.

Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem.
Remember equal number of variables and independent equations ensures a solution.

For any positive integer x, the 2-height of x is defined to be the greatest non-negative integer n such that 2^n is a factor of x. If k and m are positive integers, is the 2-height of k greater than the 2-height of m?

1. k>m
2. k/m is an even integer

2-height is a unique type of question in gmat math. In other words, 40=(2^3)5, therefore the 2 height of 40 is 3 while 16=2^4 thus the 2 height of 16 is 4. Therefore, greater number does not guarantee greater 2 height.

In the original condition, we have 2 variables (k,m) and we need 2 equations to match the number of variables and equations. Since there is 1 each in 1) and 2), there is high probability that C is the answer. Using both 1) & 2), from k/m=2integer we get k=(2integer)m, therefore k always has 2 on top of m. Thus the 2height of k is always 1 greater than that of m, and C becomes the answer. The key question is the integer question, therefore we should apply common mistake type 4(A) and therefore using just 2) already gives us the answer. 2 height of k is 1 greater than that of m. Therefore the answer is B.

Normally for cases where we need 2 more equation, such as original conditions with 2 variable, or 3 variables and 1 equation, or 4 variables and 2 equations, we have 1 equation each in both 1) and 2). Therefore C has a high chance of being the answer, which is why we attempt to solve the question using 1) and 2) together. Here, there is 70% chance that C is the answer, while E has 25% chance. These two are the key questions. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since C is most likely to be the answer according to DS definition, we solve the question assuming C would be our answer hence using ) and 2) together. (It saves us time). Obviously there may be cases where the answer is A, B, D or E.

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