Question list

1. Let n be an positive integer that is not a power of 2.
Which set of consecutive integers is n the sum of?

Question # 1

Let n be an positive integer that is not a power of 2.
Which set of consecutive integers is n the sum of?

Since n is not a power of 2, n is a multiple of a power of
2.

Let d be the largest odd positive integer that divides n.

n/d is a power of 2.

n/d = 2**k, for some positive integer k.

n is the sum of d consecutive integers, starting with
2**k - (d-1)/2.

Proof:
The average of a set of consecutive integers is the same as
the average of the first and last.

The first of the consecutive integers is, as already stated,
( 2**k - (d-1)/2 ).
The last of these consecutive integers is this first
consecutive integer, plus (d-1).

( 2**k - (d-1)/2) + (d-1) = (2**k + (d-1)/2

The average of the first and last is
( ( 2**k - (d-1)/2) + (2**k + (d-1)/2 ) ) /2
= ( 2**k + 2**k) /2 = 2**k

The sum of d positive integers, whose average is 2**k,
is d * 2**k = n.
this proves that n is the sum of d consecutive integers,
starting with 2**k - (d-1)/2.

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