How many sums
Number Theory
Level
2
consider a number k = (2^1024)*(10!)
how many different ways are there to write k as a sum of * two or more positive consecutive whole numbers *
choose from one of the given answers
10
11
5
29
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1 solution
apology and thanks : I sincerely apologize for my mistakes in the last problem "sums and sums". I thank all the very insightful members of the brilliant community for pointing out its many shortcomings. In this problem, I hope to resolve those issues. I am currently quite an amateur in the math community and your replies have helped me quite a considerable amount I apologize in advance for any defect in this problem.
solution : consider a number c which is equal to a sum of positive whole numbers starting from n and adding up d terms
c = n+n+1+n+2+...+n+d
c = dn+(sum of whole natural no.s till d)
sum of whole natural numbers = (d^2+d)/2
c = dn+((d^2+d)/2)
now we can factor the d to get c = d(n+((d+1)/2))
so from this, we can gather that d has to be odd and divide c meaning it has to be an odd factor of d
now counting the odd factors of k we get that there should be 30 solutions and taking out the trivial solution k we get there should be 29 non-trivial solutions to this problem