Three consecutive integers (guide 3, pg. 139)

[AndrejH128]

Hi everyone,

I am struggeling with the explanation of question no.6 on pg.139 in the word problem guide (5th edition).
The question goes as follows:

If x,y,z are consecutive integers, is x+y+z divisible by 3?

The answer in the guide says yes because "For any odd number of integers the sum of those integers is divisible by the number of integers"

But in my opinion this is only true if the numbers in the set don't change signs.
So for example while 1+2+3 or -5+-6+-7 are divisible by 3, -1+0+1 is not.
Since this detail is not mentioned in the question the correct answer would be no.
Or am i missing something here?

[Sage Pearce-Higgins]

I'm glad that I can clear this up. To quote from Chapter 10 of the Word Problems Strategy Guide: "For any set of consecutive integers with an ODD number of items, the sum of all the integers is ALWAYS a multiple of the number of items." This is true.

If you take your case -1, 0, 1, then the sum of these numbers is 0. This is divisible by 3. Think: what does 'divisible by 3' mean anyway? It means that when I divide it by 3, I get an integer result with no remainder. In this case 0/3 = 0, an integer result with no remainder. This makes the interesting result that 0 is divisible by everything.