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

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BTGmoderatorDC: Moderator; Posts: 7187; Joined: Thu Sep 07, 2017 4:43 pmFollowed by: 23 members

by BTGmoderatorDC » Tue Jan 22, 2019 6:27 am

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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

OA E

Source: GMAT Prep

Source: Beat The GMAT — Problem Solving | Tue Jan 22, 2019 6:27 am

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Tue Jan 22, 2019 6:29 am

BTGmoderatorDC 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

OA E

Source: GMAT Prep

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, 1Â² + 5Â² + 7Â² = 75
We want the SUM of 1 + 5 + 7, which is 13

Answer: E

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

fskilnik@GMATH: GMAT Instructor; Posts: 1449; Joined: Sat Oct 09, 2010 2:16 pmFollowed by: 33 members; Thanked: 59 times

by fskilnik@GMATH » Tue Jan 22, 2019 10:05 am

BTGmoderatorDC 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
Source: GMAT Prep

BrentÂ´s approach is (we believe) the way to go!

If we realize we need

(i) 3 odd parcels
or
(ii) 1 odd and 2 even parcels,

we reduce a bit the inspection!

Regards,
Fabio.

Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br

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by [email protected] » Tue Jan 22, 2019 10:39 am

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: E

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

Contact Rich at [email protected]

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Scott@TargetTestPrep: GMAT Instructor; Posts: 8132; Joined: Sat Apr 25, 2015 10:56 amFollowed by: 29 members; Location: Los Angeles, CA; Thanked: 43 times

by Scott@TargetTestPrep » Sun Jan 27, 2019 6:15 pm

BTGmoderatorDC 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.

Answer: E

Scott Woodbury-Stewart
Founder and CEO
[email protected]

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