Challenge

let a1,a2,........an be integers.Show that there exists integers k and r such that the sum
ak,ak+1,ak+2+.........ak+r
is divisible by n.

2 Answers

2305

Shaswata Roy ·2013-02-05 05:08:17

You must mention that 0â‰¤k<râ‰¤n
Otherwise the question becomes too easy.
And please use subscript.

However assuming 0â‰¤k<râ‰¤n.
Let S₁ = a₁ , S₂ = a₁+a₂

In general S_i = a₁+a₂+a₃+......+a_i

Let us divide each of these n sums by n.

We can either have the remainder of one of these (say S_j)=0.For such a case k=1 and r=j-1.

If none of the remainders are = 0,we must have 2 sums which have equal remainders.(Since there are n sums and n-1 possible remainders other than 0).

For such a case the difference of the 2 sums will be divisible by n.

1357

Manish Shankar ·2013-02-04 22:00:10

question is not clear, use proper subscripts

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