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805+ Level|   Algebra|      
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prince00113
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The sum of five integers is 999, while the sum of their squares is X. 01 Feb 2018, 12:22
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The sum of five integers is 999, while the sum of their squares is X. Which of the following statements is true.

A. X is never greater than 1,001,001
B. X is never less than 299794
C. X is never less than 199,601
D. X can never be a perfect square.
E. None of the above.
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C
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gracie
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The sum of five integers is 999, while the sum of their squares is X. 01 Feb 2018, 14:41
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prince00113
the sum of five integers is 999, while the sum of their squares is X. Which of the following statements is true.

A. X is never greater than 1,001,001
B. X is never less than 299794
C. X is never less than 199,601
D. X can never be a perfect square.
E. None of the above.]

999/5=199.8 arithmetic mean
let the five integers=200+200+200+200+199=999
X=199,601
X can never be less than this
C
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The sum of five integers is 999, while the sum of their squares is X. 01 Feb 2018, 19:49
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Can anyone please explain why option A is incorrect?
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The sum of five integers is 999, while the sum of their squares is X. 01 Feb 2018, 20:04
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csaluja
Can anyone please explain why option A is incorrect?

If the 5 integers are 1499,-200,-100,-100,-100(the sum is 999)
But the sum of the squares of the numbers is definitely greater than 1001001

That is the reason why A is incorrect as X is greater than 1001001

Hope that clears your confusion!
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The sum of five integers is 999, while the sum of their squares is X. 01 Feb 2018, 22:50
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prince00113
The sum of five integers is 999, while the sum of their squares is X. Which of the following statements is true.

A. X is never greater than 1,001,001
B. X is never less than 299794
C. X is never less than 199,601
D. X can never be a perfect square.
E. None of the above.


The question points towards looking for ranges..

Few points therefore..
1) B can be thrown immediately, as B constitutes the range of C..
2) Max value :-
Nothing has been said about integers being POSITIVE or NEGATIVE.
So the highest value of X can be INFINITY upwards.
Say integers are, 1000000.....n times., -10000..n times, 999, 0, 0
Here sum will be 0, but sum of squares can be anything..
Eliminate A
3) Min value :-
Min value will be when all numbers are smallest possible. And this will happen when all are close by.. here all will be ~ 200...
So Sum <5*(200)^2......<5*40000....<200000..
Here C fits in..
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The sum of five integers is 999, while the sum of their squares is X. 01 Feb 2018, 22:58
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prince00113
The sum of five integers is 999, while the sum of their squares is X. Which of the following statements is true.

A. X is never greater than 1,001,001
B. X is never less than 299794
C. X is never less than 199,601
D. X can never be a perfect square.
E. None of the above.

This can be best solved by taking all the numbers distributed equally.

\(a= b= c = d = e = 200\)
\(a^2 + b^2 + c^2 + d^2 + e^2 > 200,000\)
As it asks for minimum value \(X > 199, 601\) (Please note that I took the value of e = 200 just to simplify calculation, however it should be 199).
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The sum of five integers is 999, while the sum of their squares is X. 23 May 2022, 08:58
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