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GMATPrer Test 1 problem - Number Theory

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by medenica » Thu Jan 11, 2007 10:26 pm
"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."

How do you go about these problems? Even with the correct answer I don't quite see how to attack these. Anyone know?
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by Argen » Fri Jan 12, 2007 12:17 am
Can you tell me the answer?
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by medenica » Fri Jan 12, 2007 12:25 am
According to GMATPrep, the answer is 13.
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by Argen » Fri Jan 12, 2007 12:42 am
I think there is a generalized way to solve this kind of problem using spherical coordinates and trigonometry, but 75 is a small number, here's how I would solve it just by doing trial and error:

First we can write 75 = a^2 + b^2 + c^2

Then we can see that 0<a<b<c<9. So we only need to consider possible sums of 8 perfect squares. We'll start guessing what c is because then it'll be easier to guess the sum of a and b. We start with c=8, and a^2+b^2=75-64=11, which is not possible. Then we try 7. a^2+b^2=75-49=26. Now it's easy to see that a=1 and b=5.

It's not a generalized solution but I hope it helps.
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by GMAT dreamer » Fri Nov 21, 2008 6:04 pm
75 is multiple of 5, os square of 5= 25,
25 - 75 = 50; we know that square of 7= 49, rest 1, so square of 1=1

5 + 7 + 1 =13.

The trick is known the multiple of 75, then you can figure out the rest.
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by earth@work » Fri Nov 21, 2008 8:56 pm
GMAT dreamer wrote:75 is multiple of 5, os square of 5= 25,
25 - 75 = 50; we know that square of 7= 49, rest 1, so square of 1=1

5 + 7 + 1 =13.

The trick is known the multiple of 75, then you can figure out the rest.
liked this method, then i tried solving other numbers as well but somehow was not able to solve using this method: like took 29 (=16+9+4). cud anybody help me solve 29 using this shortcut method?
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by cramya » Fri Nov 21, 2008 9:50 pm
29 being prime would have only 2 factors 1 and 29 so I dont think the methid posted above can be applied here.

Sometimes its just easier to write out the squares and figure it out(as long as its not a huge number).
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by earth@work » Sat Nov 22, 2008 8:24 am
i guess so :-) thanks cramya
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