For a given integer n define 
to be the product of those integers
that are not divisible by p. We write 
in base p,
and let 
be the least positive residue of 
for each 
, so that 
. Finally we
let 
be the number of
`carries' on or beyond the jth digit, when adding m and r in base p.
and positive integers m=n+r, we have
where 
is 
except if p=2 and 
.
Now, with definitions as in Theorem 1, we have, for any given 
,
by
(20)
.
Multiplying together this congruence for each 
we get
Proposition 1. For any integer n and prime power
where 
,
we have
is defined as in Lemma 1 and each 
as in Theorem 1.
Theorem 1 then follows from dividing the equation in Proposition 1 by
the corresponding ones for m and r, and then using (19) to sort
out the exponents of 
and p.
