( 430 J 
1X. Another use of the series 1+ 3 45 +4 ete. 
Each divisible number consists of the sum of a few successive 
terms of the series. 
Example 57 =3 X 19 =17 419 +21, where 17 is the base 
of 57. 
So we can write: 
67 =14+3-45174941141348—384+51719431413-4(8-4+1)— 
5-7 --9-- 11-18 4+ (8-+-14+3)=749+4 114134 (8143-4 5)= 
TIPU B+ 1+ 2=—9I-U+ 1384154 (2+7)=14B+15+9+9 = 
W118 1517 +1 = 13-4 15-417 + (14-1) = 15 +17 +0 + +13) = 
15-+17-+194+6=17+19+ 91 
Or abbreviated: 
added remainder For large numbers with factors greatly 
1 : differing, this method is also very laborious. 
3 12 The possibility, however, is not excluded 
5 17 —15—2 that later on from the succession of some 
7 9 remainders we shall be able to decide upon 
9 
18—17=1 the divisibility or non-divisibility of a given 
13 2519 6. COE 
The decomposition G =a Xb + e discussed in § III finds an expla- 
nation in the series too. 
Example: G=57=>1+4+3+54+7+9-+4114138+8. 
Seven terms precede 8, of which 7 is the middle one, so G=7 7 -+-8. 
If we add 1 to 8, the 6 remaining terms have a mean 8, so 
that G == 6508 (Bj 
So we have successively : 
C= 7 + 8 namely 1+3+5+7+9+11+1348 
BSC AB eLe Gees ENE EEE Epen 
55e 0 A: 54+7+9+41141384(8+1+3) 
Oe Lien pa ame 71-0527] RAB AAA SAE) 
8 ox PEEL 94414 13-6444 Sabe 
