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PS - divisible question

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PS - divisible question

by ccassel » Sat Apr 02, 2011 12:39 pm
How would you explain the answer to this question? Does it have something to do with an odd prime #?

If n is a positive integer and n^2 is divisible by 72, then the largest positive integer that must divide n is:

A. 6
B. 12
C. 24
D. 36
E. 48

Thanks,
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by rohu27 » Sat Apr 02, 2011 9:30 pm
a similar problem was discused few days bck on this forum. let me try this again.
n^2 is divisible by 72, so n^2=72k (k is any inetegr- n^2 is a multiple of 72)
so n=sqrt(72k)=6*sqrt(2k)
the question asks for a +ve integer tht MUST divide n,->whtevr the value of n, it shud be divisible by this integer.
now n shud be an integer,so lets see for which values of k does n become an integer,
the least value of k for which n becomes an integer is k=2, then n =12
so 12 is the iteger which divides n always.

option B.
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by ccassel » Sun Apr 03, 2011 8:51 am
Thanks for the answer. Can someone expand on this?

To summarize..
- n is positive
- n^2 is divisible by 72
- what is the largest positive integer that must divide n?

1. If they are asking for the "largest positive integer that must divide n"; why would we use the smallest possible (integer) value of "k" to solve the question? For example, if we used "k=8" the answer would be (c)24.

Cheers,
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by rohu27 » Sun Apr 03, 2011 9:09 am
important thgn to note here is the word MUST, whatevr the value of n, this integer MUST divide it.
if we take the example you mentioned, if k=8, n woild be 24.
if 24 is the largest possible integer we are looking for, how abt the below case?
k=2,n=12. remember 24 MUST divide n, but 24 doesnt divide 12.
so we are finding the min. value of n so tht if for tht value the integer divides n, rest all wud just be multiples of tht particular n.
largest positive integer, - both 6 and 12 would divide all values of n, the larger one would be 12.

hope this helps.
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by tpr-becky » Tue Apr 05, 2011 8:38 am
The words "divisibe by" are your clue that this is about factors.

if n^2 is divisible by 72 that means that n^2 must have all the prime factors 72

72 has 2,2,2,3,3 as it's prime factors.

since it is n^2 that has these we know that n has a 2 and a 3 (by pulling out the factor pairs) - but there is a still a 2 left and thus n must have another 2 in it (because the square is divisible and thus can't be a decimal). thus n must have 2, 2, and 3 as factors (can have many more but must have these).

Thus n is divisible by 12 - n could be divisible by the larger numbers but we don't know for sure a thus the answer is B.

Best of Luck
Becky
Master GMAT Instructor
The Princeton Review
Irvine, CA
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