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 Post subject: The number 75 can be written as the sum of the squares
 Post Posted: Sat Jan 26, 2008 4:14 am 
The number 75 can be written as the sum of the squares of 3 different positive integers. What is the sum of these 3 integers?
A. 17
B. 16
C. 15
D. 14
E. 13

What's the efficient way to locate the answer?
Thanks.


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 Post subject: Re: The number 75 can be written as the sum of the squares
 Post Posted: Sun Jan 27, 2008 6:45 am 
Hei wrote:
The number 75 can be written as the sum of the squares of 3 different positive integers. What is the sum of these 3 integers?
A. 17
B. 16
C. 15
D. 14
E. 13

What's the efficient way to locate the answer?
Thanks.


Source not specified!!!. E. is however the correct answer. Can't ellaborate without knowing the source as Forum descipline.


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 Post subject:
 Post Posted: Sun Jan 27, 2008 9:27 pm 
Oh, I thought that this sub-forum was for asking questions from GMATPrep software ONLY.
No?
Anyway, it is from GMATPrep software at MBA.com.
Yes, the answer is E. I know why but I don't know how to come up with 1, 5 and 7 as the 3 numbers in an efficient way.
Thanks in advance.


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 Post subject:
 Post Posted: Mon Jan 28, 2008 5:24 am 
Offline
ManhattanGMAT Staff


Posts: 12762
hei is right: there is no need for formal attributions in this folder (everything is assumed to come from the gmatprep software). if you are posting problems here that are NOT from the gmatprep software, you will soon be afflicted with terrible diseases.

--

ok, there might be some crazy elegant way to solve this problem in ten seconds flat. but if it isn't rather obvious in retrospect, who cares? here's a better method:

consider odd / even numbers
note that squares of odds remain odd, and squares of evens remain even. therefore, the 3 numbers are either 3 odds or 1 odd and 2 evens.
notice that this observation kills choices b and d right away (nice if you're pressed for time and have to guess).

next observation: none of the individual numbers can be greater than 8 (because 9^2 = 81 by itself is already too big)

at this point, you have a very good idea of the sizes of numbers you're looking for, so just start trying numbers in an organized way.
3 odds:
sum = 13: (1, 5, 7)
sum = 15: (3, 5, 7)
sum = 17: impossible unless at least one number > 8

2 evens + odd:
sum = 13: (1, 4, 8) (3, 2, 8) (3, 4, 6) (5, 2, 6) (7, 2, 4)
sum = 15: (1, 6, 8) (3, 4, 8) (5, 2, 8) (5, 4, 6) (7, 2, 6) all of which are waywaywaywayway too big
sum = 17: (3, 6, 8) (5, 4, 8) (7, 2, 8) (7, 4, 6) all of which are waywaywaywayway too big

that isn't so bad.

...especially because you can stop as soon as you find (1, 5, 7); i.e., you're not going to have to go through anywhere close to everything in this list, unless you are extraordinarily unlucky.

--

if you're interested (and hardcore):
all squares of odds give remainder = 1 upon division by 4. (try it if you don't believe me)
all squares of evens give remainder = 0 upon division by 4.
(if you reallyreallyreally want to know, i can show you why these rules work, although the proofs will be 100% irrelevant to the gmat)
since 75 has remainder = 3 upon division by 4, it follows that all three numbers are odd. as evidenced in the lists above, that reduces your shortlist of candidates to two possibilities.


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 Post subject:
 Post Posted: Tue Jan 29, 2008 2:08 am 
Thanks a lot Ron!


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 Post subject:
 Post Posted: Thu Jan 31, 2008 11:22 am 
RPurewal wrote:
hei is right: there is no need for formal attributions in this folder (everything is assumed to come from the gmatprep software). if you are posting problems here that are NOT from the gmatprep software, you will soon be afflicted with terrible diseases.

--

ok, there might be some crazy elegant way to solve this problem in ten seconds flat. but if it isn't rather obvious in retrospect, who cares? here's a better method:

consider odd / even numbers
note that squares of odds remain odd, and squares of evens remain even. therefore, the 3 numbers are either 3 odds or 1 odd and 2 evens.
notice that this observation kills choices b and d right away (nice if you're pressed for time and have to guess).

next observation: none of the individual numbers can be greater than 8 (because 9^2 = 81 by itself is already too big)

at this point, you have a very good idea of the sizes of numbers you're looking for, so just start trying numbers in an organized way.
3 odds:
sum = 13: (1, 5, 7)
sum = 15: (3, 5, 7)
sum = 17: impossible unless at least one number > 8

2 evens + odd:
sum = 13: (1, 4, 8) (3, 2, 8) (3, 4, 6) (5, 2, 6) (7, 2, 4)
sum = 15: (1, 6, 8) (3, 4, 8) (5, 2, 8) (5, 4, 6) (7, 2, 6) all of which are waywaywaywayway too big
sum = 17: (3, 6, 8) (5, 4, 8) (7, 2, 8) (7, 4, 6) all of which are waywaywaywayway too big

that isn't so bad.

...especially because you can stop as soon as you find (1, 5, 7); i.e., you're not going to have to go through anywhere close to everything in this list, unless you are extraordinarily unlucky.

--

if you're interested (and hardcore):
all squares of odds give remainder = 1 upon division by 4. (try it if you don't believe me)
all squares of evens give remainder = 0 upon division by 4.
(if you reallyreallyreally want to know, i can show you why these rules work, although the proofs will be 100% irrelevant to the gmat)
since 75 has remainder = 3 upon division by 4, it follows that all three numbers are odd. as evidenced in the lists above, that reduces your shortlist of candidates to two possibilities.


In Mathematics what's elegant is never 'crazy' and what's 'crazy is never elegant.
The 'better' method can be surmised as :
Having elliminated the impossible, that whatever is left, however improbable has to be the correct answer!!!
Having said so, the 'better' method has its share of elegance , no doubt.


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 Post subject:
 Post Posted: Fri Feb 01, 2008 1:50 am 
Offline
ManhattanGMAT Staff


Posts: 7637
Location: San Francisco
By "crazy elegant" I think Ron meant "incredibly elegant." That's a common slang meaning for "crazy." :)

_________________
Stacey Koprince
Instructor
Director of Online Community
ManhattanGMAT


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 Post subject: Re: The number 75 can be written as the sum of the squares
 Post Posted: Mon Nov 02, 2009 6:38 pm 
Offline
Forum Guests


Posts: 65
Location: USA
Hi Ron i appreciate your explanation especially the remainder one

[editor: thanks, but the remainder one is WAY less important and WAY less useful. if you appreciate it more than you appreciate the trial-and-error / number plugging solution, then you don't have the right study priorities.
see my comments below. --ron]


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 Post subject: Re:
 Post Posted: Sat Nov 28, 2009 6:42 am 
Offline
Course Students


Posts: 88
RonPurewal wrote:
hei is right: there is no need for formal attributions in this folder (everything is assumed to come from the gmatprep software). if you are posting problems here that are NOT from the gmatprep software, you will soon be afflicted with terrible diseases.

--

ok, there might be some crazy elegant way to solve this problem in ten seconds flat. but if it isn't rather obvious in retrospect, who cares? here's a better method:

consider odd / even numbers
note that squares of odds remain odd, and squares of evens remain even. therefore, the 3 numbers are either 3 odds or 1 odd and 2 evens.
notice that this observation kills choices b and d right away (nice if you're pressed for time and have to guess).

next observation: none of the individual numbers can be greater than 8 (because 9^2 = 81 by itself is already too big)

at this point, you have a very good idea of the sizes of numbers you're looking for, so just start trying numbers in an organized way.
3 odds:
sum = 13: (1, 5, 7)
sum = 15: (3, 5, 7)
sum = 17: impossible unless at least one number > 8

2 evens + odd:
sum = 13: (1, 4, 8) (3, 2, 8) (3, 4, 6) (5, 2, 6) (7, 2, 4)
sum = 15: (1, 6, 8) (3, 4, 8) (5, 2, 8) (5, 4, 6) (7, 2, 6) all of which are waywaywaywayway too big
sum = 17: (3, 6, 8) (5, 4, 8) (7, 2, 8) (7, 4, 6) all of which are waywaywaywayway too big

that isn't so bad.

--

if you're interested (and hardcore):
all squares of odds give remainder = 1 upon division by 4. (try it if you don't believe me)
all squares of evens give remainder = 0 upon division by 4.
(if you reallyreallyreally want to know, i can show you why these rules work, although the proofs will be 100% irrelevant to the gmat)
since 75 has remainder = 3 upon division by 4, it follows that all three numbers are odd. as evidenced in the lists above, that reduces your shortlist of candidates to two possibilities.


Hi Ron, could you recommend any resources in which I can find these advanced number properties?
Thank you in advanced.


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 Post subject: Re:
 Post Posted: Sun Nov 29, 2009 7:52 pm 
Offline
Students


Posts: 8
In the last section -
Any specific reason you have "divided by 4" to come to a conclusion that all numbers should be odd?


Thanks
Pritesh


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 Post subject: Re: The number 75 can be written as the sum of the squares
 Post Posted: Mon Nov 30, 2009 7:02 am 
Offline
Students


Posts: 28
Another way is approach this is to list the perfect squares less than 75:

1 4 9 16 25 36 49 64

Could the sum involve 64? The sum of the other two squares would have to be 11: impossible.

How about 49? The sum of the other two squares would have to be 26= 25 + 1

Thus 75= 7^2 + 5^2 +1^2


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 Post subject: Re: Re:
 Post Posted: Thu Dec 24, 2009 6:48 pm 
Offline
ManhattanGMAT Staff


Posts: 818
pritesh.suvarna wrote:
In the last section -
Any specific reason you have "divided by 4" to come to a conclusion that all numbers should be odd?


When you add numbers together, you can also add their remainders when dividing them by a common number. For example, 14 divided by 6 is 2 R 2, and 25 divided by 6 is 4 R 1. So when we take 39 divided by 6, the remainder will be 3.

Because Ron stated that perfect squares of odd numbers have remainder of 1 when divided by 4, and perfect squares of even numbers divide evenly into 4, we want to find out the remainder when 75 is divided by 4 to determine if our three numbers are all odd, or if we have one odd and two evens.

Hope that makes sense.

_________________
Ben Ku
Instructor
ManhattanGMAT


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 Post subject: Re: The number 75 can be written as the sum of the squares
 Post Posted: Tue Jul 27, 2010 1:36 pm 
Offline
Course Students


Posts: 125
elegant indeed.


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 Post subject: Re: Re:
 Post Posted: Thu Aug 05, 2010 8:21 am 
Offline
ManhattanGMAT Staff


Posts: 12762
pritesh.suvarna wrote:
In the last section -
Any specific reason you have "divided by 4" to come to a conclusion that all numbers should be odd?


Thanks
Pritesh


well, notice that i prefaced that part with "if you're interested (and hardcore)". i would not expect very many people to come up with a solution like that (or even to understand it, really), unless they had previously undertaken an impressively extensive study of number theory.

those are inherent properties of remainders upon division by 4 -- when you take the square of an odd integer and divide it by 4, you get a remainder of 1, and when you take the square of an even integer and divide it by 4, you get a remainder of 0.
there aren't any direct hints about this in the problem; hence, the "hardcore" introduction.

in other words, it's really not much worth worrying about (unless you have an unusually deep knowledge of number theory); this problem is perfectly accessible if you just throw numbers into the equation until you get it to work.

_________________
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 Post subject: Re: The number 75 can be written as the sum of the squares
 Post Posted: Thu Aug 05, 2010 8:24 am 
Offline
ManhattanGMAT Staff


Posts: 12762
by the way, people, the "remainders upon division by 4" solution is MUCH MUCH MUCH less important than the solution by pure brute force (i.e., just throwing possibilities into the equation until it works).
not even close.

DO NOT IGNORE THE "TRIAL & ERROR" SOLUTION, AND DO NOT BECOME UNDULY OBSESSED WITH THE ADVANCED NUMBER THEORY.
the chance that you will see another problem on which you'll be able to apply the "remainders upon division by 4" method is pretty much zero.
the chance that you will see another problem on which you'll be able to use trial and error is very, very, very high.

priorities, people; priorities.

_________________
Pueden hacerle preguntas a Ron en castellano
Potete fare domande a Ron in italiano
On peut poser des questions ã Ron en français
Voit esittää kysymyksiä Ron:lle myös suomeksi

Un bon vêtement, c'est un passeport pour le bonheur.
– Yves Saint-Laurent


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