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

tagged by: swerve

This topic has 4 expert replies and 1 member reply

### Top Member

swerve Master | Next Rank: 500 Posts
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#### The number 75 can be written as the sum of the squares of...

Wed Mar 07, 2018 10:07 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

The OA is E.

I guessed this right - but is there an algebraic way to do this PS question? Please, can any expert explain it? I need your help. Thanks.

### GMAT/MBA Expert

Brent@GMATPrepNow GMAT Instructor
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Wed Mar 07, 2018 10:13 am
swerve 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
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

Cheers,
Brent

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Brent@GMATPrepNow GMAT Instructor
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Wed Mar 07, 2018 10:13 am
swerve wrote:
I guessed this right - but is there an algebraic way to do this PS question? Please, can any expert explain it? I need your help. Thanks.
No, there's no straightforward algebraic solution.

Cheers,
Brent

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Rich.C@EMPOWERgmat.com Elite Legendary Member
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Thu Mar 08, 2018 4:19 pm
Hi swerve,

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).

GMAT assassins aren't born, they're made,
Rich

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### Top Member

Vincen Legendary Member
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Fri Mar 09, 2018 7:54 am
The way I solved it was as follows:

$$75=49+25+1\ =\ 7^2+5^2+1.$$ Hence, the numbers are 1, 5 and 7. Therefore, $$7+5+1=13.$$ Therefore, the answer is E=13.

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Jeff@TargetTestPrep GMAT Instructor
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Mon Mar 12, 2018 10:34 am
swerve 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 for 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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