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Math Expert V
Joined: 02 Sep 2009
Posts: 58415
The number 95 can be written as the sum of the squares of 4 different  [#permalink]

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Difficulty:   55% (hard)

Question Stats: 64% (02:20) correct 36% (02:42) wrong based on 143 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

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Math Expert V
Joined: 02 Aug 2009
Posts: 8005
Re: The number 95 can be written as the sum of the squares of 4 different  [#permalink]

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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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Re: The number 95 can be written as the sum of the squares of 4 different  [#permalink]

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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)

hence the answer is (C)
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Re: The number 95 can be written as the sum of the squares of 4 different  [#permalink]

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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!
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The number 95 can be written as the sum of the squares of 4 different  [#permalink]

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Quote:
Since the SUM of squares of these 4 integers is ODD, the SUM of these numbers will also be ODD.

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

I too am bit confused.
If a number is odd, then its square will be odd.
But odd + Odd = even.
What did I miss here?
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Math Expert V
Joined: 02 Aug 2009
Posts: 8005
Re: The number 95 can be written as the sum of the squares of 4 different  [#permalink]

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Quote:
Since the SUM of squares of these 4 integers is ODD, the SUM of these numbers will also be ODD.

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

I too am bit confused.
If a number is odd, then its square will be odd.
But odd + Odd = even.
What did I miss here?

Hi,

So you agree that a number if squared will have the same property as far as odd and even is concerned. So odd^2 is odd and even^2 is even.

If sum of squares is 95, an odd number, then the sum of these integers will also be odd, whatever be the combination of squares.

Example ..
Three are odd and one even.
That is odd^2+odd^2+odd^2+even^2=odd...
In the same breadth, the sum of integers will be odd+odd+odd+even, which will be odd.
_________________ Re: The number 95 can be written as the sum of the squares of 4 different   [#permalink] 19 May 2019, 07:19
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