In the bracketed representations we use on this website, you may notice how the first two numbers have always the same parity. For example,
Let 
.
If 
is divisible by 
,
with 
,
then 
.
We require , so that there are at least two numbers in the representation.
The converse is not true; for a counterexample,
note that 
is not divisible by 
,
so 
is not a valid representation.
If the above is divisible by it follows that it is also divisible by 
and, as a result, so is 
. Therefore
is divisible by .
By Euclid's Lemma,
is divisible by , and
It should be evident at this point that only an even power of 
could be congruent to 
modulo ,
but we will give this a detailed treatment.
By the Division Algorithm,
let 
for some integers 
and 
, where is either 
or .
Our goal is to show that 
. So far we have
But 
.
By the rules of modular arithmetic,
we then have
But we know that can only be or , and only will satisfy the congruence above.
Therefore 
, and
