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GMAT Number Theory http://www.800score.com/forum/viewtopic.php?f=3&t=91 
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Author:  Gennadiy [ Tue Oct 11, 2011 12:13 pm ] 
Post subject:  Re: GMAT Number Theory 
questioner wrote: Please explain this some again in detail. I wasnt able to understand the explanation, probably with numbers substituted. Thanks. If there is any specific step in any reasoning above, let me know, and I'll go over it again. There can be NO shortcuts for statement (2) by considering some specific values, because this statement gives us enough information to proof that n²ª is a multiple of mª for ANY positive integers. However, you can try a simpler case, let's say for a = 1. The explanation is the same, just put 1 instead of a, whenever you meet it. The question itself will become: If m and n are positive integers, is n² a multiple of m? (1) n is a multiple of m/2 (2) n is a multiple of 2m 
Author:  questioner [ Fri Nov 04, 2011 7:27 am ] 
Post subject:  Re: GMAT Number Theory 
Let n = 2, m = 5. 2m = 10 is a divisor of n or 2, but n²ª (or 4ª) is not a multiple of mª (or 5ª). 
Author:  Gennadiy [ Fri Nov 04, 2011 7:34 am ] 
Post subject:  Re: GMAT Number Theory 
Quote: 2m = 10 is a divisor of n or 2. 2m = 10 is a multiple of n = 2, while n = 2 is a divisor of 2m = 10.

Author:  questioner [ Thu Nov 10, 2011 6:06 pm ] 
Post subject:  Re: GMAT Number Theory 
In your explanation of why answer A is insufficient, you give 2 examples of why the answer is no, which is an answer. You might inlcude an example of when the answer is yes sometimes and no other times, making A insufficient. 
Author:  Gennadiy [ Thu Nov 10, 2011 6:14 pm ] 
Post subject:  Re: GMAT Number Theory 
questioner wrote: You might inlcude an example of when the answer is yes sometimes and no other times, making A insufficient. That's exactly what we do.(1) n is a multiple of m/2 Ex. 1: m = 6, n = 3. Then 3 is a multiple of 6/2 = 3. BUT 3²ª = 9ª is clearly NOT a multiplier of 6ª. > NO. Ex. 2: m = 6, n = 6. Then 6 is a multiple of 6/2 = 3. AND 6²ª is clearly a multiplier of 6ª. > YES. 
Author:  dave [ Thu Apr 12, 2012 2:20 am ] 
Post subject:  Re: GMAT Number Theory 
Can you expand on the second point? I understand why if 2m is a divisor of n, then m is a divisor of n – but how do we get from there to understanding that n²ª is a multiple of m²ª? 
Author:  Gennadiy [ Thu Apr 12, 2012 2:44 am ] 
Post subject:  Re: GMAT Number Theory 
Quote: …, then m is a divisor of n – but how do we get from there to understanding that n²ª is a multiple of m²ª? We know that n is a multiple of m (in other words n is divisible by m).So n² is a multiple of m², because n² = n × n m² = m × m AND n²/m² = (n/m) × (n/m) is an integer, because each factor is an integer. n³ is a multiple of m³ … n²ª is a multiple of m²ª 
Author:  radz1806 [ Tue Aug 21, 2012 3:17 am ] 
Post subject:  Re: GMAT Number Theory 
A way to answer this question would be statement 1) first statement needs to be an integer as n is a multiple of m/2 so if you write it as n/(m/2) = 2(n/m) and this needs to be an integer for it to be an integer we have two options: thus n/m can be 1/2 > not an integer NO or n/m can be 1,2,3,4,.......> integers yes thus Not sufficient statement 2) it states that n/2m is an integer now separate it as (1/2) * (n/m) and this is an integer thus for this to be an integer naturally (n/m) has to be an integer which cancels out the 2 in the denominator> yes only Thus statement 2 is sufficient as if n is multiple of m then n^2 will also be a multiple of m 
Author:  Gennadiy [ Wed Aug 22, 2012 11:22 am ] 
Post subject:  Re: GMAT Number Theory 
Quote: statement 1) First statement is not a number. However you can say that n, m/2 and n / (m/2) must be integers, according to definitions of divisibility and a multiple. first statement needs to be an integer as n is a multiple of m/2 Quote: so if you write it as n/(m/2) = 2(n/m) and this needs to be an integer for it to be an integer we have two options: thus n/m can be 1/2 > not an integer NO or n/m can be 1,2,3,4,.......> integers yes thus Not sufficient n/m can be a fraction, while n²ª will be a multiple of mª . Take, for example, n = 2 and m = 4. n/m = 1/2 – a fraction n²ª = 4ª is divisible by mª = 4ª Quote: statement 2) This reasoning is correct. For formality, just add the last step.it states that n/2m is an integer now separate it as (1/2) * (n/m) and this is an integer thus for this to be an integer naturally (n/m) has to be an integer which cancels out the 2 in the denominator> yes only Thus statement 2 is sufficient as if n is multiple of m then n^2 will also be a multiple of m n² is a multiple of m, so (n²)ª is a multiple of (m)ª . 
Author:  fedana [ Wed Oct 31, 2012 5:07 pm ] 
Post subject:  Re: GMAT Number Theory 
On the GMAT, would the number 7 be considered a multiple of the number 3.5? I am wondering, whether when picking numbers for this problem, I could have picked n = 7 and m = 7 (since n is multiple of m/2, and 7 is (or not?) a multiple of 3.5) 
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