If the average of a set of consecutive integers is an integer, then the total number of integers in the set will always be odd.
ie, if there are 5 consecutive integers, only then their average will be an integer, if there were 4 integers, the avg will be a decimal.
Now, the average of consecutive integers will always have equal number of integers below it as above it. Thus, say A is the avg, and A has x number of integers below it, and x number of integers above it.
so total number of integers in the set = A + 2x
Now Even + odd = odd
So if A is odd, adding it to 2x will always give an odd integer, ie, the sum of all the integers will be odd.
coming back to understanding your logic
1) I understood that if the avg of n consequtive numbers is an integer then the average should be odd
2) and i agree that there will be equal number of integers on both sides of the average(x on each side)
3) but in your explanation you said "So if A is odd, adding it to 2x will always give an odd integer, ie, the sum of all the integers will be odd."
I thought A+ 2x is just number of integers but you seem to conclude it as Sum of the numbers, irrespective of whether either side of 'A' has any number of evens or odds.
I got the point using numbers but just want to follow your logic through the end