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



sive numbers, there is also a definite sequence and recuiTence in the sums of the 

 digits which express them ; and the results of these inquiries, with the requisite 

 demonstrations, I will now, as briefly as possible, lay before the Society. 



Prop. I. 

 K m and n are prime to one another, a m cannot equal h n, unless a and b be 

 equimultiples of M and m respectively. For, if am =6 /?,-=-. But by hyp. - is a 

 fraction in its lowest terms, therefore b=pm, and a=pn. 



Prop. II. 

 If N=P . n^\+r^ . n being the local value of the notation, and P . w— 1 being the 

 greatest multiple of n^, which is less than N ; then r„, is the ultimate sirai of the 

 digits of N. 



Let N=a + 6n + CTO^ + rfn', &C. 

 N , r , T't~. T a + b + c, &c. 



=^ = 6 + C.M+l + rf.M*+W+l+ &C +. 4 



»— 1 n—\ 



N=jB.»j — l + a + 6 + c, &c. = p.n — \ + r^ 

 Again, let r^=q.n-l + r„, where r„ is the simi of the digits of r^, or the 

 second sum of the digits of N. 



Then 'i^—p+q.n—l + r^. Let this operation be continued till *•„ becomes a 

 single digit, we have N=P . ti—i + rm, where i-„, is the ultimate sum of the digits 

 ofN. 



Ex. In our notation m=io, and »»-l=9. 

 Let N = 567434=63068 x 9 +2 



here 1st sum =29 

 2d do. =11 

 3d do. = 2 



CoK. K r =«-l, then N is a multiple of »-l. And, conversely, if N be a 

 multiple of w— 1, r„,=«— 1. 



Prop. III. 

 If a be a number prime to w-i; and p, q be two numbers, whose diflference is 

 neither n-l, nor a mizltiple of n-\, then p a and q a cannot have the same ulti- 

 mate sum. 



If possible let pa=mM^l + r and ga=^)n^.n~l + r, and let s = q-p, then 

 sa-q a-i)a=ttt^ — m + n-\; but by hyp. a is prime to n—1, and s is neither «— 1, 

 nor a multiple of it; therefore, by Prop. I. sa cannot equal m^-m^n-\, and 

 therefore j) a and q a cannot have same ultimate sum. 



