# The value of a given carry, when adding modulo p

## (The Proof of (19))

**(19)**

** Proof:**
Given integers * n* and *m*, we take *r=n-m*.
Define
if there is a `carry' in the *j*th digit when we add *m* and *r* in base *p*; otherwise let
(including ).
We observed, in the proof of Kummer's Theorem that, for each integer
,

Therefore, if we let
be the least residues, in absolute
value, of , respectively, so that times the
left side of (19), plus equals *n-m-r = 0*. However,
and (19) follows.