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For any positive integer x, the 2height of x is defined to be the
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26 Oct 2015, 09:58
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For any positive integer x, the 2height of x is defined to be the greatest nonnegative integer n such that 2^n is a factor of x. If k and m are positive integers, is the 2height of k greater than the 2height of m? (1) k > m (2) k/m is an even integer. Kudos for a correct solution.
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Re: For any positive integer x, the 2height of x is defined to be the
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27 Oct 2015, 02:16
Bunuel wrote: For any positive integer x, the 2height of x is defined to be the greatest nonnegative integer n such that 2^n is a factor of x. If k and m are positive integers, is the 2height of k greater than the 2height of m?
(1) k > m (2) m/k is an even integer.
Kudos for a correct solution. In simple words, 2height is just the number of 2s in a positive integer x. So if x = 40, its 2height will be 3 because 40 = 8*5 = 2*2*2*5 If x = 15, its 2height is 0 because there are no 2s in 15. and so on. So, an even number will have a 2height of at least 1. An odd number will have a 2height of 0. "is the 2height of k greater than the 2height of m?" means "Does k have more 2s than m?" (1) k > m k could have more 2s than m or it could have fewer 2s than m. For example, if k = 4 and m = 3, k has two 2s while m has none. If k = 11 and m = 8, k has no 2s while m has 3. Not sufficient. (2) m/k is an even integer. When m is divided by k, you get an integer. So m has all factors of k and they get cancelled out and you are left with an integer. Also, the leftover integer is even so m has at least one 2 more than k. So 2height of m is certainly more than the 2height of k. So we can answer the question that k does not have more 2s than m. Answer (B)
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Re: For any positive integer x, the 2height of x is defined to be the
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23 Dec 2015, 22:15
Although Karishma's approach (as always) is spot on and is an optimal way to solve such questions as it directly simplifies the layer of complexity in a logical way, when I attempted this question for first time I solved by plugging numbers.
Statement 1: K>M, if K=6 and M=4, then answer is No but if K=8 and M=4, the answer is Yes. Hence, insufficient. Statement 2: M/K is an even integer, which means that M has all the factors of K (including powers of 2, i.e. 2height of K) AND it has at least an extra 2 (because M/K results in an even integer, i.e. M=K*(an even integer)). Therefore, 2height of K will always be smaller than 2height of M. Hence, sufficient.
Answer  B




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Re: For any positive integer x, the 2height of x is defined to be the
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26 Oct 2015, 19:29
I think the answer is A.
I could derive a equation from the given information as
{2^(2K) / K} > {2^(2M) / M}...? ( Not Sure )
Statement 1 K>M, Now substituting values according to this will keep the stem true.
Hence Sufficient.
Statement 2 Not sufficient.
Hope my approach is fine.



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Re: For any positive integer x, the 2height of x is defined to be the
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23 Dec 2015, 23:01
Bunuel wrote: For any positive integer x, the 2height of x is defined to be the greatest nonnegative integer n such that 2^n is a factor of x. If k and m are positive integers, is the 2height of k greater than the 2height of m?
(1) k > m (2) m/k is an even integer.
Kudos for a correct solution. The Q basically asks us if the power of 2 is greater for k when compared to m.. 1) k>m.. not suff as k could be an odd number and m an even number, which would mean the 2height of k < the 2height of m.. and say k=32 and x=2.. the 2height of k >the 2height of m insuff 2)m/k is an even integer... this clearly shows that m is some even multiple of k.. and thus it will always be true that the 2height of k < the 2height of m.. suff B
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Re: For any positive integer x, the 2height of x is defined to be the
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27 Dec 2015, 12:23
Bunuel wrote: For any positive integer x, the 2height of x is defined to be the greatest nonnegative integer n such that 2^n is a factor of x. If k and m are positive integers, is the 2height of k greater than the 2height of m?
(1) k > m (2) m/k is an even integer.
Kudos for a correct solution. My approach was simple picking numbers case 1) K>M K=40, M=39: 2height = 2^5=32 => same height K= 65 M=40: 2height K= 6, 2height for M=5 => different INSUFFICIENT Case 2) M/K is an even integer This means M is either 0 times of N or M is atleast twice of N, since M, N are +ve Integer 0 is invalid, which means M is atleast twice of N, which leads 2height of M will always be at least 1 greater than 2height of N. Sufficient.



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Re: For any positive integer x, the 2height of x is defined to be the
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28 Jul 2016, 07:55
In the OG 2016, the "(2)" is not "m/k" but rather "k/m". Answer explanation p.337: "Given that k/m is an even integer, it follows that k/m = 2n for some integer n, or k = 2mn. This implies that the 2height of k is at least one more than the 2height of m; SUFFICIENT."



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Re: For any positive integer x, the 2height of x is defined to be the
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28 Jul 2016, 08:08
Bunuel wrote: For any positive integer x, the 2height of x is defined to be the greatest nonnegative integer n such that 2^n is a factor of x. If k and m are positive integers, is the 2height of k greater than the 2height of m?
(1) k > m (2) m/k is an even integer.
Kudos for a correct solution. Lets understand the question first 2height of any number x= power of 2 in factorial of XLets check the statements: Statement1: K>m So can we surely say k has greater power of 2 than m? No. Lets see why. It can be true for k=2^3;m=2^2 it can be false for k=(2^3)*5 and m=(2^3)*3 Notice both above examples fulfil the statement 1 but give uncertain results. Hence InsufficientStatement 2: m/k is even integer since k and m are positive integer. this means m/k=2 times some integer or m has more power of 2 than k. hance 2height of m will be greater than2 height of k hence sufficient. Ans: BKudos if you like my solution



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Re: For any positive integer x, the 2height of x is defined to be the
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27 Aug 2016, 05:00
Can someone please explain what is being asked? Preferably with algebraic equation.
Thanks



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Re: For any positive integer x, the 2height of x is defined to be the
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27 Aug 2016, 05:13
sashlev wrote: Can someone please explain what is being asked? Preferably with algebraic equation.
Thanks Questions has explained the meaning of 2height of x. It says the 2height of x is defined to be the greatest non negative integer n such that 2^n is a factor of x. or we have n>=0 such that 2^n is a factor of x. Say x=2, so we know that 2^1 ihe greatest factor of x. So, its 2height will be 1. Similarly, if we say x =28, we have 2^2 as the greatest factor in x, so n = 2, and so on. Now, I hope the question is clear. Moving on to the statements : Statement 1 : k>m. Say K =5 and m is 4. in this case 2height of m is greater than that is k. But say k=8 and m=4, its the other way. Hence, insufficient. Statement 2 : m/k is an even integer. for m/k to be an integer, we must have k<=m. Also, since it is even,we can say m must have 2 in it. Or I can directly say m is a multiple of k such that m is even. => 2 height if m will always be greater than that of k. Take any values of m and k such that the above equation i satisfied you will always get the result. hence, B is insufficient
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Re: For any positive integer x, the 2height of x is defined to be the
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06 Jun 2017, 10:06
VeritasPrepKarishma wrote: Bunuel wrote: For any positive integer x, the 2height of x is defined to be the greatest nonnegative integer n such that 2^n is a factor of x. If k and m are positive integers, is the 2height of k greater than the 2height of m?
(1) k > m (2) m/k is an even integer.
Kudos for a correct solution. In simple words, 2height is just the number of 2s in a positive integer x. So if x = 40, its 2height will be 3 because 40 = 8*5 = 2*2*2*5 If x = 15, its 2height is 0 because there are no 2s in 15. and so on. So, an even number will have a 2height of at least 1. An odd number will have a 2height of 0. "is the 2height of k greater than the 2height of m?" means "Does k have more 2s than m?" (1) k > m k could have more 2s than m or it could have fewer 2s than m. For example, if k = 4 and m = 3, k has two 2s while m has none. If k = 11 and m = 8, k has no 2s while m has 3. Not sufficient. (2) m/k is an even integer. When m is divided by k, you get an integer. So m has all factors of k and they get cancelled out and you are left with an integer. Also, the leftover integer is even so m has at least one 2 more than k. So 2height of m is certainly more than the 2height of k. So we can answer the question that k does not have more 2s than m. Answer (B) Thank you very much for this . I didn't even comprehend the question so I certainly had no chance of answering correctly. I was a bit demoralized till I read your great explanation. Does the GMAT use height to represent that amount of times a prime shows up when doing prime factorization often?



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Re: For any positive integer x, the 2height of x is defined to be the
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11 Jun 2017, 06:18
for 1, for 3>2, 2heights are 0 and 1 4>2, 2heights are 2 and 1 insufficient
for 2, m/k is even integer so m will have atleast 2height 1 more than k sufficient
Answer B



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Re: For any positive integer x, the 2height of x is defined to be the
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16 Jun 2017, 06:42
NanaA wrote: VeritasPrepKarishma wrote: Bunuel wrote: For any positive integer x, the 2height of x is defined to be the greatest nonnegative integer n such that 2^n is a factor of x. If k and m are positive integers, is the 2height of k greater than the 2height of m?
(1) k > m (2) m/k is an even integer.
Kudos for a correct solution. In simple words, 2height is just the number of 2s in a positive integer x. So if x = 40, its 2height will be 3 because 40 = 8*5 = 2*2*2*5 If x = 15, its 2height is 0 because there are no 2s in 15. and so on. So, an even number will have a 2height of at least 1. An odd number will have a 2height of 0. "is the 2height of k greater than the 2height of m?" means "Does k have more 2s than m?" (1) k > m k could have more 2s than m or it could have fewer 2s than m. For example, if k = 4 and m = 3, k has two 2s while m has none. If k = 11 and m = 8, k has no 2s while m has 3. Not sufficient. (2) m/k is an even integer. When m is divided by k, you get an integer. So m has all factors of k and they get cancelled out and you are left with an integer. Also, the leftover integer is even so m has at least one 2 more than k. So 2height of m is certainly more than the 2height of k. So we can answer the question that k does not have more 2s than m. Answer (B) Thank you very much for this . I didn't even comprehend the question so I certainly had no chance of answering correctly. I was a bit demoralized till I read your great explanation. Does the GMAT use height to represent that amount of times a prime shows up when doing prime factorization often? I am surprised that nobody has answered this question. I am a native English speaker, and someone who has spent a substantial amount of time studying GMAT quantatitive questions and I have never encountered this terminology ("height") when referring to factors of a number. For this reason, I had no idea how to answer the question. I can only hope this obscure phraseology was used to increase the difficulty of the question.



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For any positive integer x, the 2height of x is defined to be the
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17 Jun 2017, 01:04
ilovefrankee wrote: Thank you very much for this . I didn't even comprehend the question so I certainly had no chance of answering correctly. I was a bit demoralized till I read your great explanation. Does the GMAT use height to represent that amount of times a prime shows up when doing prime factorization often?
Quote: I am surprised that nobody has answered this question. I am a native English speaker, and someone who has spent a substantial amount of time studying GMAT quantatitive questions and I have never encountered this terminology ("height") when referring to factors of a number. For this reason, I had no idea how to answer the question.
I can only hope this obscure phraseology was used to increase the difficulty of the question. This is not standard terminology so its no surprise that you haven't come across it before. It is a user defined function for this question only. The question gives exactly how to calculate 2Height for a positive integer. There could be another question which could define 2Height as "the number of 2s that will add up to give that integer" etc. Yes, user defined functions can increase the difficulty level of the question though note that the concepts are still very simple. The questions only look more difficult than they actually are.
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Re: For any positive integer x, the 2height of x is defined to be the
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17 Jun 2017, 02:09
I'm confused. In DS questions, I treat both statements provided as objectively true. Therefore, if we find values that work with one of the statements, they cannot contradict the other statement. In this questions: (1) k > m (2) m/k is an even integer. I understand why (2) works, but if m/k is an even integer, and both variables are positive, then how can statement (1) ever be qualified as true? The only scenario in which this works is if m = 0, but that would cause the question to become invalid since the 2height if m=0 cannot be calculated. Am I missing something? Bunuel or VeritasPrepKarishma, your help would be appreciated.



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Re: For any positive integer x, the 2height of x is defined to be the
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17 Jun 2017, 04:59
bionication wrote: I'm confused. In DS questions, I treat both statements provided as objectively true. Therefore, if we find values that work with one of the statements, they cannot contradict the other statement. In this questions: (1) k > m (2) m/k is an even integer. I understand why (2) works, but if m/k is an even integer, and both variables are positive, then how can statement (1) ever be qualified as true? The only scenario in which this works is if m = 0, but that would cause the question to become invalid since the 2height if m=0 cannot be calculated. Am I missing something? Bunuel or VeritasPrepKarishma, your help would be appreciated. You are right. There was a typo in the second statement. (2) reads k/m is an even integer, not m/k is an even integer. Edited. Thank you for noticing it.
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Re: For any positive integer x, the 2height of x is defined to be the
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28 Jul 2017, 03:08
VeritasPrepKarishma wrote: Bunuel wrote: For any positive integer x, the 2height of x is defined to be the greatest nonnegative integer n such that 2^n is a factor of x. If k and m are positive integers, is the 2height of k greater than the 2height of m?
(1) k > m (2) m/k is an even integer.
Kudos for a correct solution. In simple words, 2height is just the number of 2s in a positive integer x. So if x = 40, its 2height will be 3 because 40 = 8*5 = 2*2*2*5 If x = 15, its 2height is 0 because there are no 2s in 15. and so on. So, an even number will have a 2height of at least 1. An odd number will have a 2height of 0. "is the 2height of k greater than the 2height of m?" means "Does k have more 2s than m?" (1) k > m k could have more 2s than m or it could have fewer 2s than m. For example, if k = 4 and m = 3, k has two 2s while m has none. If k = 11 and m = 8, k has no 2s while m has 3. Not sufficient. (2) m/k is an even integer. When m is divided by k, you get an integer. So m has all factors of k and they get cancelled out and you are left with an integer. Also, the leftover integer is even so m has at least one 2 more than k. So 2height of m is certainly more than the 2height of k. So we can answer the question that k does not have more 2s than m. Answer (B) Your solution definitely needs my time to stop and say 'Thank You!'.



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Re: For any positive integer x, the 2height of x is defined to be the
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10 Nov 2017, 19:01
0 is also even integer, so what if k/m is 0?
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Re: For any positive integer x, the 2height of x is defined to be the
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10 Nov 2017, 19:32
Cheryn wrote: 0 is also even integer, so what if k/m is 0? Hi, always read the entire q thoroughly so as not to miss out on minor details. It is given m and k are positive integers so they cannot be 0
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Re: For any positive integer x, the 2height of x is defined to be the
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17 Nov 2017, 05:43
VeritasPrepKarishma wrote: Bunuel wrote: For any positive integer x, the 2height of x is defined to be the greatest nonnegative integer n such that 2^n is a factor of x. If k and m are positive integers, is the 2height of k greater than the 2height of m?
(1) k > m (2) m/k is an even integer.
Kudos for a correct solution. In simple words, 2height is just the number of 2s in a positive integer x. So if x = 40, its 2height will be 3 because 40 = 8*5 = 2*2*2*5 If x = 15, its 2height is 0 because there are no 2s in 15. and so on. So, an even number will have a 2height of at least 1. An odd number will have a 2height of 0. "is the 2height of k greater than the 2height of m?" means "Does k have more 2s than m?" (1) k > m k could have more 2s than m or it could have fewer 2s than m. For example, if k = 4 and m = 3, k has two 2s while m has none. If k = 11 and m = 8, k has no 2s while m has 3. Not sufficient. (2) m/k is an even integer. When m is divided by k, you get an integer. So m has all factors of k and they get cancelled out and you are left with an integer. Also, the leftover integer is even so m has at least one 2 more than k. So 2height of m is certainly more than the 2height of k. So we can answer the question that k does not have more 2s than m. Answer (B) Responding to a pm: Quote: In your explanation, you mention:
"So, an even number will have a 2height of at least 1."
Isn't 0 an even number with a 2height (as defined in the question) of 0? That wouldn't affect the answer to the question since we know both k and m to be positive integers, but I just wanted to be clear conceptually with respect to the statement you made about even numbers in general.
Note that the function is defined for positive integers only. Hence, whatever general statements we make regarding this function, they are made for positive integers only.
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