Last visit was: 22 Jun 2025, 04:59 It is currently 22 Jun 2025, 04:59
Close
GMAT Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
Close
Request Expert Reply
Confirm Cancel
User avatar
enigma123
Joined: 25 Jun 2011
Last visit: 16 Mar 2016
Posts: 392
Own Kudos:
Given Kudos: 217
Status:Finally Done. Admitted in Kellogg for 2015 intake
Location: United Kingdom
Concentration: International Business, Strategy
GMAT 1: 730 Q49 V45
GPA: 2.9
WE:Information Technology (Consulting)
GMAT 1: 730 Q49 V45
Posts: 392
Kudos: 18,594
 [256]
21
Kudos
Add Kudos
233
Bookmarks
Bookmark this Post
Most Helpful Reply
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 22 Jun 2025
Posts: 102,227
Own Kudos:
Given Kudos: 93,968
Products:
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 102,227
Kudos: 734,458
 [150]
71
Kudos
Add Kudos
79
Bookmarks
Bookmark this Post
avatar
PareshGmat
Joined: 27 Dec 2012
Last visit: 10 Jul 2016
Posts: 1,538
Own Kudos:
7,860
 [109]
Given Kudos: 193
Status:The Best Or Nothing
Location: India
Concentration: General Management, Technology
WE:Information Technology (Computer Software)
Posts: 1,538
Kudos: 7,860
 [109]
87
Kudos
Add Kudos
21
Bookmarks
Bookmark this Post
User avatar
BrentGMATPrepNow
User avatar
Major Poster
Joined: 12 Sep 2015
Last visit: 13 May 2024
Posts: 6,757
Own Kudos:
33,890
 [3]
Given Kudos: 799
Location: Canada
Expert
Expert reply
Posts: 6,757
Kudos: 33,890
 [3]
2
Kudos
Add Kudos
1
Bookmarks
Bookmark this Post
enigma123
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

We're looking for 3 DIFFERENT squares that add to 75

Here are the only squares we need to consider: 1, 4, 9, 16, 25, 36, 49, 64
Can you find 3 that add to 75?
After some fiddling, we may notice that 1 + 25 + 49
In other words, 1² + 5² + 7² = 75
We want the SUM of 1 + 5 + 7, which is 13

Answer: E

Cheers,
Brent
General Discussion
avatar
sandeeepsharma
Joined: 26 Apr 2011
Last visit: 12 Apr 2017
Posts: 189
Own Kudos:
47
 [12]
Given Kudos: 14
Products:
Posts: 189
Kudos: 47
 [12]
8
Kudos
Add Kudos
4
Bookmarks
Bookmark this Post
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
User avatar
mikemcgarry
User avatar
Magoosh GMAT Instructor
Joined: 28 Dec 2011
Last visit: 06 Aug 2018
Posts: 4,480
Own Kudos:
30,036
 [6]
Given Kudos: 130
Expert
Expert reply
Posts: 4,480
Kudos: 30,036
 [6]
5
Kudos
Add Kudos
1
Bookmarks
Bookmark this Post
Hi, there. I'm happy to give my 2 cents on this. :)

First of all, this is a relatively difficult question for GMAT math. You would only see a question of this difficulty if you were already answering almost all of the math questions correctly.

I can't really suggest much other than trial-and-error. Since 25 is a factor of 75, I figured it might make sense if 25 were one of the three squares. That means the other two squares would have to add up to 50. Well, conveniently, 7^2 = 49, so 7^2 + 1^2 = 50, and 7^2 + 1^2 + 5^2 = 75. The three numbers are 1, 5, and 7, and those have a sum of 13. Answer = E.

Here, I was really just following my intuition for numbers, which is really just a notch above pure guess-and-check. I don't know the source of this question, but the real GMAT tends to give questions that admit of either a methodical approach or an elegant solution, whereas this more or less requires some guess & check, some poking around in the dark. That's just not the style of questions that the GMAC dishes out.

Please let me know if you have any questions on what I've said here.

Mike :)
User avatar
enigma123
Joined: 25 Jun 2011
Last visit: 16 Mar 2016
Posts: 392
Own Kudos:
Given Kudos: 217
Status:Finally Done. Admitted in Kellogg for 2015 intake
Location: United Kingdom
Concentration: International Business, Strategy
GMAT 1: 730 Q49 V45
GPA: 2.9
WE:Information Technology (Consulting)
GMAT 1: 730 Q49 V45
Posts: 392
Kudos: 18,594
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Thanks for a very clear explanation Bunuel.
User avatar
budablasta
Joined: 23 Feb 2012
Last visit: 19 Apr 2017
Posts: 195
Own Kudos:
89
 [5]
Given Kudos: 22
Location: India
Concentration: Finance, Entrepreneurship
Schools: Said
GMAT 1: 710 Q44 V44
GPA: 2.9
WE:Marketing (Computer Software)
Schools: Said
GMAT 1: 710 Q44 V44
Posts: 195
Kudos: 89
 [5]
4
Kudos
Add Kudos
1
Bookmarks
Bookmark this Post
Bunuel
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.
User avatar
calreg11
Joined: 27 Oct 2011
Last visit: 07 Mar 2013
Posts: 84
Own Kudos:
1,033
 [1]
Given Kudos: 4
Location: United States
Concentration: Finance, Strategy
GPA: 3.7
WE:Account Management (Consumer Packaged Goods)
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
thanks for the solution... was hoping for another solution other than just brute calculation. bunel you rock tho.
avatar
GMAT17325
Joined: 07 Jun 2014
Last visit: 09 Mar 2021
Posts: 69
Own Kudos:
Given Kudos: 42
Posts: 69
Kudos: 9
Kudos
Add Kudos
Bookmarks
Bookmark this Post
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.
User avatar
swanidhi
Joined: 11 Oct 2013
Last visit: 14 Jul 2023
Posts: 70
Own Kudos:
310
 [2]
Given Kudos: 137
Concentration: Marketing, General Management
GMAT 1: 600 Q41 V31
GMAT 1: 600 Q41 V31
Posts: 70
Kudos: 310
 [2]
Kudos
Add Kudos
2
Bookmarks
Bookmark this Post
enigma123
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
User avatar
EMPOWERgmatRichC
User avatar
Major Poster
Joined: 19 Dec 2014
Last visit: 31 Dec 2023
Posts: 21,789
Own Kudos:
12,445
 [6]
Given Kudos: 450
Status:GMAT Assassin/Co-Founder
Affiliations: EMPOWERgmat
Location: United States (CA)
GMAT 1: 800 Q51 V49
GRE 1: Q170 V170
Expert
Expert reply
GMAT 1: 800 Q51 V49
GRE 1: Q170 V170
Posts: 21,789
Kudos: 12,445
 [6]
5
Kudos
Add Kudos
1
Bookmarks
Bookmark this Post
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:
GMAT assassins aren't born, they're made,
Rich
User avatar
ScottTargetTestPrep
User avatar
Target Test Prep Representative
Joined: 14 Oct 2015
Last visit: 21 Jun 2025
Posts: 20,985
Own Kudos:
26,036
 [3]
Given Kudos: 293
Status:Founder & CEO
Affiliations: Target Test Prep
Location: United States (CA)
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 20,985
Kudos: 26,036
 [3]
1
Kudos
Add Kudos
2
Bookmarks
Bookmark this Post
enigma123
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.

Answer: E
avatar
MTLGMAT514
Joined: 05 Feb 2019
Last visit: 10 Mar 2020
Posts: 1
Given Kudos: 1
Location: United States
Schools: HSG SIM "22
Schools: HSG SIM "22
Posts: 1
Kudos: 0
Kudos
Add Kudos
Bookmarks
Bookmark this Post
I got this question on a practice test I took via GMAC.

Oddly enough, if you prime factorize 75 (15->5 and 3 and 5), you will get 5+5+3= 13. Is that a coincidence or is there actually something behind it?

I know that the question is not asking for this, but I found it a bit strange. Any insight on this would be helpful! Thank you
User avatar
EMPOWERgmatRichC
User avatar
Major Poster
Joined: 19 Dec 2014
Last visit: 31 Dec 2023
Posts: 21,789
Own Kudos:
Given Kudos: 450
Status:GMAT Assassin/Co-Founder
Affiliations: EMPOWERgmat
Location: United States (CA)
GMAT 1: 800 Q51 V49
GRE 1: Q170 V170
Expert
Expert reply
GMAT 1: 800 Q51 V49
GRE 1: Q170 V170
Posts: 21,789
Kudos: 12,445
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Hi MTLGMAT514,

Since you can't actually get to MOST integers by adding up three squares of integers, this result is just a coincidence.

GMAT assassins aren't born, they're made,
Rich
avatar
maxraxstax
Joined: 24 Jan 2020
Last visit: 16 Mar 2021
Posts: 8
Given Kudos: 3
Location: United States (NY)
GMAT 1: 710 Q47 V41
GPA: 3.51
GMAT 1: 710 Q47 V41
Posts: 8
Kudos: 0
Kudos
Add Kudos
Bookmarks
Bookmark this Post
mikemcgarry
Hi, there. I'm happy to give my 2 cents on this. :)

First of all, this is a relatively difficult question for GMAT math. You would only see a question of this difficulty if you were already answering almost all of the math questions correctly.

I can't really suggest much other than trial-and-error. Since 25 is a factor of 75, I figured it might make sense if 25 were one of the three squares. That means the other two squares would have to add up to 50. Well, conveniently, 7^2 = 49, so 7^2 + 1^2 = 50, and 7^2 + 1^2 + 5^2 = 75. The three numbers are 1, 5, and 7, and those have a sum of 13. Answer = E.

Here, I was really just following my intuition for numbers, which is really just a notch above pure guess-and-check. I don't know the source of this question, but the real GMAT tends to give questions that admit of either a methodical approach or an elegant solution, whereas this more or less requires some guess & check, some poking around in the dark. That's just not the style of questions that the GMAC dishes out.

Please let me know if you have any questions on what I've said here.

Mike :)

newish here - how come this is labeled sub 600 if it's tough? I found this on a official practice gmat
User avatar
EMPOWERgmatRichC
User avatar
Major Poster
Joined: 19 Dec 2014
Last visit: 31 Dec 2023
Posts: 21,789
Own Kudos:
12,445
 [1]
Given Kudos: 450
Status:GMAT Assassin/Co-Founder
Affiliations: EMPOWERgmat
Location: United States (CA)
GMAT 1: 800 Q51 V49
GRE 1: Q170 V170
Expert
Expert reply
GMAT 1: 800 Q51 V49
GRE 1: Q170 V170
Posts: 21,789
Kudos: 12,445
 [1]
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Hi maxraxstax,

As far as the 'knowledge' and the 'skills' needed to answer this question are concerned, this is not a particularly difficult question (at its core, you're just adding some 1-digit and 2-digit numbers together - and that's low level Arithmetic). As such, the sub-600 'label' is appropriate.

GMAT assassins aren't born, they're made,
Rich
User avatar
rsrobin864
Joined: 21 Aug 2020
Last visit: 10 Jan 2025
Posts: 66
Own Kudos:
Given Kudos: 60
Location: India
Products:
Posts: 66
Kudos: 76
Kudos
Add Kudos
Bookmarks
Bookmark this Post
­Thanks Bunel for the approach
avatar
ManifestDreamMBA
Joined: 17 Sep 2024
Last visit: 22 Jun 2025
Posts: 854
Own Kudos:
Given Kudos: 188
Products:
Posts: 854
Kudos: 534
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Another Method:

75 = 3*25 = 3 * 5^2
If all numbers could have been same, then sum would have been 5*3 = 15. Given this, the sum cannot be more than 15, since for any number more than 5, the other number must be less than 5.
This eliminates A,B,C.

Given the square of values add to odd number, all the values must add to odd. So only 13 works
enigma123
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
Moderators:
Math Expert
102227 posts
PS Forum Moderator
654 posts