Prove that $4$ divides $n$

Question

Let $a_1$,$a_2$,$a_3$,.......,$a_n$ be $n$ such that each $a_i$ either $1$ or $-1$.If

$a_1 a_2 a_3 a_4+a_2 a_3 a_4 a_5+......+a_n a_1 a_2 a_3=0$,

then prove that $4$ divides $n$.

I tried this for small n and also observed different specific cases such as $a_1=1$ $a_2=-1$ $a_3=1$ $a_4=-1$.... etc.

But I am unable to circumvent anyway. Any kind of hints and full answers will be appreciated. Thank you all in advance.

I'm fairly sure that this has been handled on our site recently. Not sure whether the duplicate had contest-math tag. — Jyrki Lahtonen, Aug 11 '15 at 10:08
Found it - by simply Googling "a_1a_2a_3a_4 site:math.stackexchange.com" — Bart Michels, Aug 11 '15 at 10:14
@JyrkiLahtonen Actually it's more like a proof-explanation request than a duplicate. That's why I just posted a link in the comments without marking as a duplicate. — Bart Michels, Aug 11 '15 at 10:26
No accusations of copying were made! I simply closed this, because it has been asked and answered before. Happens often enough here, believe me :-). Closing duplicates is a site rule. — Jyrki Lahtonen, Aug 11 '15 at 10:31

score 4 · Accepted Answer · answered Aug 11 '15 at 10:06

Hint: When you change $a_r$ from $-1$ to $+1$ you change the value of four of the summands. Each summand which changes changes in value by $2$ - either from $+1$ to $-1$ or from $-1$ to $+1$. If you change all the $-1$ terms $a_r$ to $+1$ what happens to the sum?

Prove that $4$ divides $n$

1 Answers1