522 Mr. W. H. Walenn on Unitation. 



(10-S)»-^ + (lO-8) n ~ 2 b + (10-S)"- 3 c + . . . + (10-S) 2 s 

 + (10-8) 1 /+(10-8) w+(10-S)- ] y+(10~8)- 2 w 



+ ...+_(l'0--P)-*2 



has the same remainder to 8 as the given number has, n being 

 the number of integer digits in the given number, and m being 

 the number of digits on the right-hand side of the decimal 

 point. 



The simplest form of the theorem from which the above ex- 

 pansion can be made is the statement as put forth in the first 

 paper on the subject*, namely : — u that if t be the tens' and u the 

 units' digit of a two-figure number, and 8 be any integer less 

 than 10, then 



has the same remainder to 8 as 10t-\-u." The following proof 

 of this simplest statement is also applicable to the expanded 

 forms of the theorem. By multiplication, 



(lQ-S)t + u = lQt-St + u=z(lOt-hu)-St', 



but the latter expression only differs from (10/ -fw) by an exact 

 number of times 6\ Further, if the digits be s, t, u, the same 

 is true of (10 — S) 2 ,? + (10 — 8)t + u, and so on ; for each time 10 

 occurs as a factor in any term, it must be treated in the way 

 above indicated. 



12. Another proof of the general theorem may be given, by 

 induction. It is well known that the addition of the two-figure 

 digits (excepting 99) in the nine-times multiplication table 

 always gives nine. Tf a method of obtaining a similar property 

 in other numerals were discovered, the remainders to division by 

 the said numerals would be found without using the operation 

 of division ; for if the process left any unit, and if the unit were 

 not already less than the divisor, it would be reducible to a 

 quantity less than the divisor, and therefore to the remainder 

 required, by simple subtraction. In the case of the nine-times 

 table, the following are instances: — 9.3 = 27, and 2 + 7 = 9; 

 and 9 .3 + 4 = 31, and 3 + 1=4. 



To oblige the multiplication table of any digit to constantly 

 yield the digit itself (by a process analogous to casting out 

 the nines), the units will certainly have to be added; what 

 other operation the remainder of the digits must be subjected 

 to is the problem to be solved. The operation of addition 

 (when it does not involve factors) is clearly not sufficient ; for, 

 as an operation, it admits of remainders that are no function 

 of the divisor, except in the case of the nine-times Table. 

 This consideration, combined with the inspection of, say, the 

 * Phil. Mag. S. 4. vol. xxxvi. p. 346. 



