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Problem Of The Week 3

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Find the sum of all positive integers n such that x,y,z are positive factors of n-1, x>y>z and x+y+z=n

a) 13               b)  31                  c) 44                d)   105                   e) none of these


Written by Implex

August 27, 2008 at 12:41 pm

One Response

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  1. Solution !
    factors are of the form (n-1)/2 , (n-1)/3 ,(n-1)/4, (n-1)/5….
    They might be integer or not….but if integers they are the factors in descending order…

    Highest possible sum= (n-1)/2 + (n-1)/3 + (n-1)/4 =13(n-1)/12 > (n-1)
    Second highest = (n-1)/2 + (n-1)/3 + (n-1)/5= 31(n-1)/30
    Next highest sum = (n-1)/2 + (n-1)/4 + (n-1)/5 = 19(n-1)/20 n=13
    31(n-1)/30=n => n=31

    so n=13,31
    hence sum 31+13=44
    option c)


    August 28, 2008 at 11:05 am

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