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# The number 95 can be written as the sum of the squares of 4 different

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Math Expert
Joined: 02 Sep 2009
Posts: 43380
The number 95 can be written as the sum of the squares of 4 different [#permalink]

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27 May 2016, 08:00
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55% (hard)

Question Stats:

67% (01:54) correct 33% (02:10) wrong based on 72 sessions

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The number 95 can be written as the sum of the squares of 4 different positive integers in two ways. Of the following, which can be the sum of these 4 integers?

(A) 9
(B) 10
(C) 15
(D) 18
(E) 20
[Reveal] Spoiler: OA

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Math Expert
Joined: 02 Aug 2009
Posts: 5545
Re: The number 95 can be written as the sum of the squares of 4 different [#permalink]

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27 May 2016, 08:50
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Bunuel wrote:
The number 95 can be written as the sum of the squares of 4 different positive integers in two ways. Of the following, which can be the sum of these 4 integers?

(A) 9
(B) 10
(C) 15
(D) 18
(E) 20

Hi,

I will give you two ways..

1) CHOICES
EASY, if you understand properties of numbers and apply them correctly....
CRITICAL info - Since the SUM of squares of these 4 integers is ODD, the SUM of these numbers will also be ODD..
ONLY 9 and 15 are ODD in choices...

The minimum SUM of 4 different positive integers = 1+2+3+4 = 10
so SUM>9....

ONLY 15 left
ans C

2) the biggest possible integer in these four integer will be 9 , as 10^2 goes above 95...
$$1^2 = 1.................2^2 = 4................3^2 = 9...................4^2 =16......................5^2 = 25...............6^2=36.............7^2= 49.............8^2=64..............9^2 = 81...$$
Since SUM is ODD, try combinations of 3 odd numbers and 1 even. OR 3 even and 1 odd......
now we can try some combinations and we will get our answer after some tries...
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Absolute modulus :http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html

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Joined: 07 Mar 2016
Posts: 16
Location: Indonesia
GPA: 3.06
Re: The number 95 can be written as the sum of the squares of 4 different [#permalink]

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28 May 2016, 02:53
2
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Bunuel wrote:
The number 95 can be written as the sum of the squares of 4 different positive integers in two ways. Of the following, which can be the sum of these 4 integers?

(A) 9
(B) 10
(C) 15
(D) 18
(E) 20

The only possible sum for 4 different squares of an integer with total sum number is 95 are :

49 + 36 + 9 + 1 = 95 which are the square from numbers : 7, 6, 3, 1 (total sum number is 17)

or

81 + 9 + 4 + 1 = 95 which are the square from numbers : 9, 3, 2, 1 (total sum number is 15)

Manager
Joined: 17 May 2016
Posts: 67
GMAT 1: 720 Q49 V39
GPA: 3.7
WE: Analyst (Investment Banking)
Re: The number 95 can be written as the sum of the squares of 4 different [#permalink]

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31 May 2016, 12:47
chetan2u wrote:
CRITICAL info - Since the SUM of squares of these 4 integers is ODD, the SUM of these numbers will also be ODD..
ONLY 9 and 15 are ODD in choices...

Thank you for such a time-saving shortcut. I only have one question - where do we derive the aforementioned information from? Thank you!
_________________

My debrief: Self-study: How to improve from 620(Q39,V36) to 720(Q49,V39) in 25 days!

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Joined: 09 Sep 2013
Posts: 14143
Re: The number 95 can be written as the sum of the squares of 4 different [#permalink]

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13 Sep 2017, 10:20
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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Re: The number 95 can be written as the sum of the squares of 4 different   [#permalink] 13 Sep 2017, 10:20
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