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# The number 75 can be written as the sum of the squares

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Manager
Joined: 05 Aug 2008
Posts: 89
Schools: McCombs Class of 2012
The number 75 can be written as the sum of the squares [#permalink]

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24 Dec 2008, 17:23
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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

Attachment:

Q1.JPG [ 15.13 KiB | Viewed 844 times ]
SVP
Joined: 07 Nov 2007
Posts: 1761
Location: New York
Re: Sum of squares [#permalink]

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24 Dec 2008, 21:22
smarinov wrote:
Attachment:
Q1.JPG

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

Ans = 1+5+7=13

E
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Manager
Joined: 05 Aug 2008
Posts: 89
Schools: McCombs Class of 2012
Re: Sum of squares [#permalink]

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24 Dec 2008, 21:26
x2suresh,

that is the correct answer, was there an approach you took to get to this answer, or did you just plug numbers and used your intuition/knowledge to see that 5^2 + 7^2 + 1 would give you 75?

thanks
Intern
Joined: 26 Dec 2008
Posts: 1
Re: Sum of squares [#permalink]

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26 Dec 2008, 02:24
1 4 9 16 25 36 49

75 is an odd number, so:
a) 2 of 3 our numbers must be even and one odd (3 cases: 4,16; 4;36; 16;36)
b) 3 of our numbers must be odd (4 cases: 1,9,25; 1,25,49; 1,9,49; 9,25,49)
Intern
Joined: 26 Dec 2008
Posts: 10
Schools: Thunderbird
Re: Sum of squares [#permalink]

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27 Dec 2008, 00:28
2, 3, 8 also works. There has to be a formula behind this.

Can anyone come up with it?
Intern
Joined: 27 Nov 2008
Posts: 5
Re: Sum of squares [#permalink]

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27 Dec 2008, 04:21
Both 7,5,1 and 8,3,2 add up to 13.
So it doesn't really matter which set you come up with.
SVP
Joined: 07 Nov 2007
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Location: New York
Re: Sum of squares [#permalink]

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27 Dec 2008, 07:30
amopk wrote:
Both 7,5,1 and 8,3,2 add up to 13.
So it doesn't really matter which set you come up with.

Anyway 64+9+4 = 77 not equal to 75
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Re: Sum of squares [#permalink]

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27 Dec 2008, 10:35
...ugh...
GMAT Tutor
Joined: 24 Jun 2008
Posts: 1346
Re: Sum of squares [#permalink]

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27 Dec 2008, 20:02
It's possibly overkill to do any analysis here, since there aren't many possibilities regardless, but if 75 were a larger number, we could at least cut down on the number of cases by doing a bit of math. As wkeeper points out, there are two cases:

* two evens, one odd
* three odds

The first case is mathematically impossible, because 75 doesn't have the right remainder when you divide by 4. Even squares are always multiples of 4, and odd squares are always 1 greater than multiples of 4 (actually, than multiples of 8, but we don't need that here). So, if you add two even squares and an odd square, you'll get a number with a remainder of 1 when you divide by 4. That's not true of 75, so we must be adding three odd squares here, and after we see that, there really aren't many possibilities to check.
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Re: Sum of squares   [#permalink] 27 Dec 2008, 20:02
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# The number 75 can be written as the sum of the squares

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