Question

How many integers 1≤N≤1000 can be written both as the sum of 26 consecutive integers and as the sum of 13 consecutive integers?

Details and assumptions
The consecutive integers are allowed to be a mix of negative integers, 0 and positive integers (as long as they are consecutive).

Answer 1

26 consecutive numbers... 13 odd numbers, 13 even numbers
13 consecutive numbers... 7 odd numbers, 6 even numbers (has to be so, or it would be even, and it can't be even because you can't write an even number as the sum of 26 consecutive numbers)

Answer 2

26 consecutive numbers: \(a_1,a_2,a_3\dots,a_{26}\).
actually: \(a_2=a_1+1, a_3=a_1+2\), etc.
so the consecutive numbers are \(a_1, a_1+1,a_1+2,...,a_1+25\)
try to see what the sum of these numbers look like and how the sum changes if you go from \((a_1, a_1+1,a_1+2,...,a_1+25)\) to \((a_1+1,a_1+2,...,a_1+26)\).

Answer 3

also, what would be the smallest \(a_1\) to consider?