Casting out nines is a way to check ones arithemetic using module mathematics since the value of a number modulo 9 can be calculated by repeated summing the digits and then setting 9 equal to 0.
In[1]:= cnine[n_Integer] :=
FixedPoint[Total[IntegerDigits[#]] &, n] /. 9 -> 0
In[2]:= Mod[Range[1000], 9] == cnine /@ Range[1000]
Out[2]= True
Casting out elevens is tricker, since the repeated operation is alternatedly adding and subtracting the digits, starting from the right and adding 11 if the result is negative.
In[3]:= t[l_List] :=
If[# < 0, # + 11, #] & @
Total[(-1)^Range[0, Length[l] - 1]*Reverse[l]]
In[4]:= celev[n_Integer] := FixedPoint[t[IntegerDigits[#]] &, n]
In[5]:= Mod[Range[1000], 11] == celev /@ Range[1000]
Out[5]= True
Perhaps someone can produce a more elegant implementation.