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# The number 75 can be written as the sum of the squares of 3

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The number 75 can be written as the sum of the squares of 3 [#permalink]

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26 Dec 2011, 11:15
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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

Guys - as the OA is not provided can someone please help me and explain how to solve this?
[Reveal] Spoiler: OA

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26 Dec 2011, 11:20
The numbers are 5,7, and 1. The answer is Option (E).
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27 Dec 2011, 03:53
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I think this question has been discussed earlier also.
squares of natural numbers, which are below 75,are 1,4,9,16,25,36,49,64
1+25+49=75 is the only option
so numbers are 1,5,7
sum =13

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19 Jan 2012, 19:19
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enigma123 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?
17
16
15
14
13

Guys - as the OA is not provided can someone please help me and explain how to solve this?

Responding to a pm.

I think brute force with some common sense should be used to solve this problem.

Write down all perfect squares less than 75: 1, 4, 9, 16, 25, 36, 49, 64.

Now, 75 should be the sum of 3 of those 8 numbers. Also to simplify a little bit trial and error, we can notice that as 75 is an odd numbers then either all three numbers must be odd (odd+odd+odd=odd) OR two must be even and one odd (even+even+odd=odd).

We can find that 75 equals to 1+25+49=1^2+5^2+7^2=75 --> 1+5+7=13.

Similar question to practice: the-number-90-can-be-written-as-the-sum-of-the-squares-of-128396.html
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Re: The number 75 can be written as the sum of the squares of 3 [#permalink]

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24 Jan 2012, 18:37
Thanks for a very clear explanation Bunuel.
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12 Mar 2012, 23:59
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Bunuel wrote:
I think brute force with some common sense should be used to solve this problem.

Write down all perfect squares less than 75: 1, 4, 9, 16, 25, 36, 49, 64.

Now, 75 should be the sum of 3 of those 8 numbers. Also to simplify a little bit trial and error, we can notice that as 75 is an odd numbers then either all three numbers must be odd (odd+odd+odd=odd) OR two must be even and one odd (even+even+odd=odd).

Very elegant solution. I got this question in my GMAT Prep 1. I also figured that it would require brute force. But Bunuel's point about 75 being an odd number and therefore being a sum of either odd-odd-odd or odd-even-even adds an elegant touch to the brute force approach.
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Re: The number 75 can be written as the sum of the squares of 3 [#permalink]

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18 Apr 2012, 20:11
thanks for the solution... was hoping for another solution other than just brute calculation. bunel you rock tho.
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Re: The number 75 can be written as the sum of the squares of 3 [#permalink]

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29 Jun 2014, 20:12
is there any way to solve this one faster? I find myself just trying to add the different numbers and spending 3 mins on this questions, just due to the calculations. any tricks on this one? Thanks.

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Re: The number 75 can be written as the sum of the squares of 3 [#permalink]

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08 Aug 2014, 01:23
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GMAT17325 wrote:
is there any way to solve this one faster? I find myself just trying to add the different numbers and spending 3 mins on this questions, just due to the calculations. any tricks on this one? Thanks.

75 can be expressed as addition of squares of same integer as follows

$$75 = 5^2 + 5^2 + 5^2$$

Leaving one square term as it is; try to search for other options

$$75 = 5^2 + 50$$

Closest is 1 & 49

$$75 = 5^2 + 1 + 49$$

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

Answer = 5+1+7 = 13 = E
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Re: The number 75 can be written as the sum of the squares of 3 [#permalink]

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09 Aug 2014, 08:31
I Seriously cannot believe I got this one wrong. I forgot to take into consideration square of 1 is 1 and guessed an answer. Still managed to get a score of 48 on this GMATprep, undeserved if I commit such silly mistakes. Good method Paresh
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Re: The number 75 can be written as the sum of the squares of 3 [#permalink]

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30 Apr 2015, 02:02
enigma123 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

Guys - as the OA is not provided can someone please help me and explain how to solve this?

just pick numbers

most of questions of math test us ability to pick numbers.

not easy one
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Re: The number 75 can be written as the sum of the squares of 3 [#permalink]

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12 Jun 2015, 00:34
Yeah I misread this question and thought it was asking for factors so I had 5 5 3 which also add up to 13

Oh God.
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Jerome: *$&#(*%&(*#%& Kudos [?]: 9 [0], given: 18 Intern Joined: 13 Aug 2014 Posts: 38 Kudos [?]: 11 [1], given: 20 GMAT 1: 750 Q51 V39 The number 75 can be written as the sum of the squares of 3 [#permalink] ### Show Tags 10 Jul 2015, 16:48 1 This post received KUDOS FatRiverPuff wrote: Yeah I misread this question and thought it was asking for factors so I had 5 5 3 which also add up to 13 Oh God. Um...the sum of factors of 75 = 1 + 3 + 5 + 15 + 25 + 75. You just got catastrophically lucky! Oh God indeed. Kudos [?]: 11 [1], given: 20 Current Student Joined: 11 Oct 2013 Posts: 121 Kudos [?]: 67 [0], given: 137 Concentration: Marketing, General Management GMAT 1: 600 Q41 V31 The number 75 can be written as the sum of the squares of 3 [#permalink] ### Show Tags 29 Nov 2015, 01:31 enigma123 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 Guys - as the OA is not provided can someone please help me and explain how to solve this? Bunuel's bruteforce using odd and even looks promising! I'd also like to add that you could eliminate some answer options using the same approach. Since 75 is an odd number. The squares must be of the form - $$Odd^2 + Odd^2 + Odd^2$$ OR $$Even^2 + Even^2 + Odd^2$$ Alternatively - $$Odd + Odd + Even$$ OR $$Even + Even + Odd$$. Thus, the sum of these numbers must be ODD. Eliminate answer option B and C. Breaking 13, 15, and 17 using further bruteforce. We get $$7^2 + 1^2 + 5^2$$. 7 + 1 + 5 = 13. Answer Option A Bunuel - Please correct me I am wrong, but if this is a general question that appears on the exam, we can conclude - Sum of squares equal to a number? * If the number is odd --> Correct answer option will be Odd. * If the number is even --> Correct answer option will be Even _________________ Its not over.. Kudos [?]: 67 [0], given: 137 EMPOWERgmat Instructor Status: GMAT Assassin/Co-Founder Affiliations: EMPOWERgmat Joined: 19 Dec 2014 Posts: 10119 Kudos [?]: 3511 [0], given: 173 Location: United States (CA) GMAT 1: 800 Q51 V49 GRE 1: 340 Q170 V170 Re: The number 75 can be written as the sum of the squares of 3 [#permalink] ### Show Tags 29 Nov 2015, 11:05 Hi All, This question can be solved with a bit of 'brute force' arithmetic, but the speed with which you solve it will likely depend on how quickly you can write everything down and how you do the work. We're asked to find the three positive integers, whose squares add up to 75. To start, we should write down the list of possible squares: 1 4 9 16 25 36 49 64 We're limited to these 8 numbers. To work efficiently, we should work from 'greatest to least'.. If we use 64, then the other two numbers have to add up to 11....but there's NO WAY to make that happen (so 64 is NOT one of the numbers we need). Next, let's try 49... then the other two numbers have to add up to 26....25 and 1 'fit', so the three numbers are 49, 25 and 1 (7, 5 and 1 --> that sum = 13). Final Answer: [Reveal] Spoiler: E GMAT assassins aren't born, they're made, Rich _________________ 760+: Learn What GMAT Assassins Do to Score at the Highest Levels Contact Rich at: Rich.C@empowergmat.com # Rich Cohen Co-Founder & GMAT Assassin Special Offer: Save$75 + GMAT Club Tests Free
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Re: The number 75 can be written as the sum of the squares of 3 [#permalink]

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29 Nov 2015, 11:14
enigma123 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

Guys - as the OA is not provided can someone please help me and explain how to solve this?

1^2 = 1
2^2 = 4
3^2 = 9
4^2 = 16
5^2 = 25
6^2 = 36
7^2 = 49
8^2 = 64

9^2 = 81
10^2 = 100

Quote:
The sum of the squares of 3 different positive integers is 75

Start from 8^2 = 64

75 - 64 = 11 ; we need 2 other squares to add upto 11 - No possibility

Then 7^2 = 49

75 - 49 = 26 ; we need 2 other squares to add upto 26 - Possibile squares are 25 , 1

So, the required numbers are 7 , 5 and 1

Hence the numbers are 1,5,7

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Re: The number 75 can be written as the sum of the squares of 3 [#permalink]

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26 Jun 2016, 22:18
enigma123 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

Guys - as the OA is not provided can someone please help me and explain how to solve this?

75 = a^2 + b^2 + c^2

1^2 = 1
2^2 = 4
3^2 = 9
4^2 = 16
5^2 = 25
6^2 = 36
7^2 = 49
8^2 = 64

We need to put in the values of a,b and c from the above list of values and see which ones satisfy.
71 = 49 + 25 + 1 = 7^2 + 5^2 + 1^2
Hence (a,b,c) = (7,5,1)
a+b+c = 7+5+1 = 13

Correct Option: E
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Re: The number 75 can be written as the sum of the squares of 3 [#permalink]

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26 Jul 2017, 16:43
enigma123 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

If the sum of 3 different perfect squares is 75, each must be less than 75. So, we want to start by writing out the perfect squares that are less than 75:

1, 4, 9, 16, 25, 36, 49, 64

In this problem, there is no shortcut to determining which 3 squares add up to 75; however, it is strategic to start with the largest one, 64, and move down the list if it doesn’t work:

64 + 11 = 75

There is no way 11 can be written as a sum of two squares. So, we move down to 49:

49 + 26 = 75

We see that 26 can be written as the sum of 25 and 1; that is:

49 + 25 + 1 = 75

We have found the three perfect squares that sum to 75. In other words:

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

The sum of 7, 5, and 1 is 7 + 5 + 1 = 13.

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Re: The number 75 can be written as the sum of the squares of 3   [#permalink] 26 Jul 2017, 16:43
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