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# How many different sets of positive square integers, each

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Manager
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How many different sets of positive square integers, each [#permalink]

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03 Nov 2009, 22:28
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How many different sets of positive square integers, each greater than 1, add up to 75?

(A) 1
(B) 4
(C) 7
(D) 11
(E) 13

Thanks.

Rohit
[Reveal] Spoiler: OA
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04 Nov 2009, 05:33
highest square integer <75 is 64.
64 +9, 1 set
25 + 25 + 25 is same square number repeated thrice.
so IMO A.
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Manager
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04 Nov 2009, 06:39
can't we consider { 25, 25 , 25 } and {64 , 9} two sets here.??.
it is not mentioned that numbers in sets are different but sets are unique..
if we don not consider {25 , 25 , 25} then obviously 1 is the answer..there are no combination equal to four so all can be eliminated...
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04 Nov 2009, 08:48
I go for b) 4.

{25,25,25} is one. Now each of these 25 can replaced with a (19,6) combo - so we get 3 more sets by replacing one 25, then two 25s, then three 25s.

That way the the total is 4, as below:
{25,25,25}
{9,16,25,25}
{9,16,9,16,25)
{9,16,9,16,9.16}
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04 Nov 2009, 09:33
Good thinking 4test1
Yes I will agree that the numbers can be same,but the sets have to be different.
In that case -

{25,25,25}
{9,16,25,25}
{9,16,9,16,25)
{9,16,9,16,9.16}

{64,9}
{16,16,16,16,9}
{4,4,4,4;4,4,4,4;4,4,4,4;4,4,4,4;9}

Total 7.
IMO C.
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04 Nov 2009, 10:16
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Ah right - amit2k9 fleshed it out properly.

1. {25,25,25}
2. {9,16,25,25}
3. {9,16,9,16,25)
4. {9,16,9,16,9,16}

And in 2,3,4 above, we can replace each 16 by four 4s which gives 6 more.

So a total of 4+6 = 10.

Why is everyone listing (64,9) - doesn't this add up to 73 while the question asks for 75?
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10 Nov 2009, 06:05
hi again,
sorry my last post is that bad formatted (i sent i also accidently)
here again are the different sets:

#1 25 25 25
#2 16 9 25 25
#3 16 9 16 9 25
#4 16 9 16 9 16 9
#5 25 16 9 9 4 4 4 4
#6 25 25 9 4 4 4 4
#7 25 9 4 4 4 4 9 4 4 4 4
#8 9 4 4 4 4 9 4 4 4 4 9 4 4 4 4
#9 9 4 4 4 4 16 9 16 9
#10 9 4 4 4 4 9 4 4 4 4 16 9
#11 9 9 9 9 9 9 4 4 4 9
#12 36 9 9 4 4 4 9
#13 49 9 4 4 9

each line should sum up to 75, and as far as i see there is noone count double
m
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11 Nov 2009, 05:20
Brilliant attemp by all you guys..

but still i would go with A ,
bcoz according to the definition of 'set' it is the collection of distinct objects.. so we cant consider duplicated in sets.. thats why IMO its only one set : {64,9}
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11 Nov 2009, 10:29
Hey 4test1, thanks for correcting me,
I made the same mistake as ppl did in above posts, however my point of concern was that according to the definition of Set, we cannot consider duplicates in it.

but i m wondering if we consider this restriction , then i don't think we aren't getting any set that fulfills the criteria...

can naybody solve my doubt??

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11 Nov 2009, 10:57
The sets are different (distinct), not the numbers within the set.
So imo 25,25,25 is valid.
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11 Nov 2009, 20:50
4 sets

25,25,25,
16,9,25,25
16,9,16,9,25
16,9,16,9,16,9

Not sure why everyone's listing 64 and 9 !!!
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12 Nov 2009, 03:02
hi!
@ pleonasm: what about the other sets e.g. {4;4;4;4;9;25;25}
(see my earlier post)
this set fullfills also the requirements, doesn't it?
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18 Nov 2009, 07:46
exi wrote:
hi again,
sorry my last post is that bad formatted (i sent i also accidently)
here again are the different sets:

#1 25 25 25
#2 16 9 25 25
#3 16 9 16 9 25
#4 16 9 16 9 16 9
#5 25 16 9 9 4 4 4 4
#6 25 25 9 4 4 4 4
#7 25 9 4 4 4 4 9 4 4 4 4
#8 9 4 4 4 4 9 4 4 4 4 9 4 4 4 4
#9 9 4 4 4 4 16 9 16 9
#10 9 4 4 4 4 9 4 4 4 4 16 9
#11 9 9 9 9 9 9 4 4 4 9
#12 36 9 9 4 4 4 9
#13 49 9 4 4 9

each line should sum up to 75, and as far as i see there is noone count double
m

I agree with the solution given by exi. In fact, there is an explanation here, but they seem to have missed the set #13 above.

That said, 13 (number of different sets) is not even an answer choice. Are we missing anything here? And what is the OA and if any explanation available for it please?

Update: It seems this is a weekly challenge question from Manhattan GMAT, published in GMatters - November 11, 2009 and they posted it here too. I'd be interested in seeing the solution that they come up with (today, I believe).
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19 Nov 2009, 07:10
gnet wrote:
It seems this is a weekly challenge question from Manhattan GMAT, published in GMatters - November 11, 2009 and they posted it here too. I'd be interested in seeing the solution that they come up with (today, I believe).

Just to let you all know that Manhattan GMAT did eventually correct their answer choices to include 13 as the option E for this question. 13 is the correct answer, here is the solution they have posted (same as given by exi).
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30 Nov 2009, 23:05
is it a GMAT/OG Q?
did not quite understand it

the positive squares i am getting are
4,9,16,25,36,49,64

now what?
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21 Feb 2010, 06:36
Hi guys,
1) I don't believe it is GMAT problem. Q50/Q51 problems are tricky problems but not time consuming at least to such extent.
2) set is a unique collection of objects. In other words, {25,25,25} is equal {25} under math definition of set. GMAT uses word "list" or "data set" to avoid confusion. By the way, I don't remember any ambiguous GMAT math problem.
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Re: sum of positive integers [#permalink]

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21 Jan 2012, 02:52
Bunuel wrote:
Merging similar topics.

Why Bunuel?

See Walker's post above, where he explains that you won't see such time-consuming questions on the GMAT.
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How many different sets of positive square integers = 75 ? [#permalink]

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17 May 2012, 18:45
How many different sets of positive square integers, each greater than 1, add up to 75?

(A) 1
(B) 4
(C) 7
(D) 10
(E) 12
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Re: How many different sets of positive square integers = 75 ? [#permalink]

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17 May 2012, 23:11
carcass wrote:
How many different sets of positive square integers, each greater than 1, add up to 75?

(A) 1
(B) 4
(C) 7
(D) 10
(E) 12

Merging similar topics.
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Re: How many different sets of positive square integers, each [#permalink]

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19 May 2012, 06:26
I read the rest of the topic.............such kind of questions implant in you a lot of not necessary clutter.

Thanks Mod, by the way.
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Re: How many different sets of positive square integers, each   [#permalink] 19 May 2012, 06:26

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