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Making integer an expression (Posted on 2009-03-20) Difficulty: 2 of 5
Find all positive integers n which make the expression below an integer:

(25/2 + (625/4 - n) ) + (25/2 - (625/4 - n) )

See The Solution Submitted by pcbouhid    
Rating: 3.0000 (1 votes)

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not gross enough | Comment 6 of 12 |

I suggested only the single value (144, 7) for what I assume was also the reason Jer adduced, that no part of the expression should be an imaginary/complex term.  This would mean that n should not be greater than 156.25, and hence 144 would be the unique solution.  Since Charlie thinks otherwise, invoking the title "just math" in support, I suppose we must see what pcbouhid had in mind: the phrase "find all positive integers n..." suggests more than one, but it also suggests a finite number and not an infinitude of answers.  Further, why should he have limited n to only positive integers unless he thought the problem needed that restriction?  Only he can say.  My days of formal math are decades behind me, since I first marveled at deMoivre's theorem.

With Charlie's interpretation, may I point out, we would not only need to have definition for the square root of a complex number, but also of the square root of the square root of a complex number if the second major factor had a real component less than 12.5.  In that case we may not be able to dodge the problem by mere "cancellation" of  the complexes. 

I propose the original solution, however gross if may appear.


  Posted by ed bottemiller on 2009-03-20 16:31:00
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