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

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

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New post 27 May 2016, 09:00
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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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Re: The number 95 can be written as the sum of the squares of 4 different [#permalink]

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New post 27 May 2016, 09:50
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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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Re: The number 95 can be written as the sum of the squares of 4 different [#permalink]

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New post 28 May 2016, 03:53
2
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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New post 31 May 2016, 13:47
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!
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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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New post 13 Sep 2017, 11:20
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Re: The number 95 can be written as the sum of the squares of 4 different   [#permalink] 13 Sep 2017, 11:20
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