90 BISHOP TERROT ON THE SUMS OF THE DIGITS OF NUMBERS. 



Ex. 2. But let the number whose multiples are taken have a common divisor 

 with 9 («-l) as 6. 



The series is 6, 12, 18, 24, 30, 36, 42, 48, 54, 60. 



Sums . . 6, 3, 9, 6, 3, 9, &c. 

 Where the sums recur at every thu-d term, because 6 and 9 have a common 



divisor 3, and '^ = 3, that is ^ =3. (Prop. V.) 



Ex. 3. The recm-rence of the sums, according to Prop. V., may be more 

 strikingly illustrated, if we use a notation whose root is 13, and, consequently, 

 n^rT= 12. If we express tlie successive multiples of 6 in this notation, we must 

 adopt three additional characters for 10, 11, 12. Let these be l^,, 1,, 1^. The 

 successive multiples in this notation are, 



6, 1„, 15, 11 J, 24, 21 „, 33, 39, 42, 48, &c. 



Sums . . 6, Ij, 6, I5, 6, l^, &c., Avhere we see the sums recur after 



12 

 two terms, because « — 1=12, and -7r=2. 



o 



Prop. VIII. 

 If n be even, and «-i consecutive terms of an arithmetic series be taken, the 

 ultimate sum of the digits of their aggregate is n-1. But if n be odd, the ulti- 

 mate sum will be n-1, or sum ( '^'~o"~" ) according as k the common differ- 

 ence, is even or odd. 



n — 1 



For aggregate of n-1 terms of arithmetic series = {2a + n—2.b) . ^^ 



= a.>V^l + '^:z }-^~^° . If n be even " ~^ is integer, whether b be even or odd. 



Therefore tlie latter term is a multiple of n-1, and, consequently, the whole 

 expression being a multiple of w-l, has n-1 for its ultimate sum (Prop. II., Cor.) 



But if n be odd, ^o" ^^ integer only when b is even. 



Prop. IX. 



If Ave assume as bases two numbers whose sum is s7?i-l, and take a series 

 of the successive powers of each, then of the two series expressing the sums of 

 digits of successive powers, the even terms are identical, while the odd terms are 

 complemental, that is, their sum is n-1. 



Let m + »Jj = s.n — 1, m^ = s.n—l — m 



w'= (*-.«^n^/-p(«.n^^""\?t ± /// according as /) is even or 



odd. Here every term except the last is a multiple of s.n- 1. 



Therefore if;? be even, sum of »«j= sum ofm''. 

 But if p be odd, sum of oti= sum of (Q.»-l)-sum of m =n-l- sum of « • 



