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OG16 - DS 121

This topic has 1 expert reply and 1 member reply
amina.shaikh309 Senior | Next Rank: 100 Posts Default Avatar
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OG16 - DS 121

Post Sun Jun 26, 2016 3:10 am
Elapsed Time: 00:00
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    Post Sun Jun 26, 2016 4:05 am
    Quote:
    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
    We can think of the 2-height of x is being the greatest n when x/(2^n) = integer. So another way to think of this is that the 2-height of x is the number of 2's that x contains in its prime factorization. For example, the 2-height of 16 would simply be 4, as 16 = 2^4. The 2 height of 12 would be 2 as 12 = 2^2*3.

    Statement 1: Pick some numbers. Case 1: k = 6 and m = 4.
    The 2-height of k is 1 as 6 = 2 * 3
    The 2-height of m is 2 as 4 = 2^2.
    We get a NO, the 2-height of k is not greater than the 2-height of m

    Case 2: k = 8
    m = 4
    The 2-height of k is 3 as 8 = 2^3
    The 2-height of m is 2 as 4= 2^2
    We get a YES, the 2-height of k is greater than the 2-height of m.

    Because we can get a YES and a NO, statement 1 is not sufficient.

    Statement 2:
    If k/m is an even integer, then we can reuse k = 8 and m = 4, as 8/4 = 2. So we know we can get a YES.
    Or try k = 32 and m = 8.
    2-height of k is 5, as 32 = 2^5
    2-height of m is 3, as 8 = 2^3
    Another YES

    You'll see quickly that no matter what you pick, you'll get a YES. So statement 2 is sufficient and the answer is B

    (In essence, we're trying to determine which number has more 2's. If k/m is EVEN, we can say that k/m = 2*integer. Or k = m*2*integer.
    No matter what you pick for m, you're multiplying that number by 2, so k will always contain more 2's.)

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    800_or_bust Master | Next Rank: 500 Posts Default Avatar
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    Post Sun Jun 26, 2016 11:24 am
    amina.shaikh309 wrote:
    (1) Not sufficient. Consider k = 9, m = 8. Then the 2-height of k is 0 and the 2-height of m is 3. But if k = 16, m = 8, the 2-height of k is 4 and 2-height of m is 3. Since the 2-height of k can be greater or less than the 2-height of m, such that k > m, this is insufficient.

    (2) Sufficient. Since k and m are both positive integers, this implies k = 2x*m, where x is a positive integer. Thus, it must be the case that k has at least one more factor of 2 than does m. Hence, the 2-height of k must be greater than the 2-height of m.

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