Chap. XII] CRITERIA FOR DIVISIBILITY. 339 
a=F&/nH-c/n^=F . . . is divisible by 10n±l, with a like test for 10n±3 
(replacing 1/n by 3/n), and deduced the usual tests for 9, 11, 7, 13, etc. 
A. L. Crelle^^ noted that to test XmA""-^ . . . +XiA-i-Xo for the divisor s 
we may select any integer n prime to s, take r=nA (mod s), and test 
for the divisor s. For example, if A = 10, s = 7, 10^=— 1 (mod 7), so that 
Xo — Xi-\-X2—. . . ±0:^ is to be tested for the divisor 7, where Xq, . . .are the 
three-digit components of the proposed number from right to left. Simi- 
larly for s=9, 11, 13, 17, 19. 
A. Transon^^ gave a test for the divisibility of a number by any divisor 
of 10"-n±l. 
A. Niegemann^^ noted that 354578385 is divisible by 7 since 35457 -f 
2X8385 is divisible by 7. In general if the number formed by the last m 
digits of A^ is multiplied by k, and the product is added to the number de- 
rived from N by suppressing those digits, then N is divisible by d if the 
resulting sum is divisible by d. Here k{0<k<d) is chosen so that 10'"/: — 1 
is divisible by d. Thus k = 2 if m = 4, d = 7. 
Many of the subsequent papers are listed at the end of the chapter. 
H. Wilbraham^^ considered the exponent p to which 10 belongs modulo 
m, where m is not divisible by 2 or 5. Then the decimal for 1/m has 
a period of p digits. If any number N be marked off into periods of p- 
digits each, beginning with units, so that A^ = ai + 10^a2+10^^a34- • • •, 
then ai-\-a2-]- ■ ■ ■ = N (mod m), and N is divisible by m if and only if 
«i+<J2+ • • ■ is divisible by m. 
E. B. Elliott^^ let 10'' = MD+r^,. Thus iV = 10%-h . . . +10ni+no is 
divisible by D if N ='ZfnjMD-{-'Znjrj is divisible by D. The values of the r's 
are tabulated for D = S, 7, S, 9, 11, 13, 17. 
A. Zbikowski^° noted that N = a-\-10kis divisible by 7 if k — 2a is divis- 
ible by 7. If 8 is of the form lOn+1, N = a-\-10k is divisible by 5 if A; — na is 
divisible by d ; this holds also if 5 is replaced by a divisor of a number 10n+ 1 . 
V. ZeipeP^ tests for a divisor h by use of nh = 10d-\-l. Then 10a2+ai is 
divisible by 6 if a2 — aid is divisible by b. 
J. C. Dupain^^ noted, for use when division by p — 1 is easy, that 
N={p — 1)Q+R is divisible by p if R — Q is divisible by p. 
F. Folie^^ proved that if a, c are such that ak'^ck = mp then AB-\-C is 
divisible by the prime p = aB-{-c if Ak'=^Ck = m'p, provided a, c, k, k' are 
isjour. fur Math., 27, 1844, 125-136. 
16N0UV. Ann. Math., 4, 1845, 173-4 (cf. 81-82 by O. R.). 
i^Entwickelung u. Begrlindung neuer Gesetze iiber die Theilbarkeit der Zahlen. Jahresber. 
Kath. Gym. Koln, 1847-8. 
i^Cambridge and Dublin Math. Jour., 6, 1851, 32. 
"The Math. Monthly (ed. Runkle), 1, 1859, 45-49. 
"oBull. ac. sc. St. Petersbourg, (3), 3, 1861, 151-3; Melanges math. astr. ac. St. P^tersbourg, 
3, 1859-66, 312. 
2iOfversigt finska vetensk. forhandl., Stockhohn, 18, 1861, 425-432. 
«Nouv. Ann. Math., (2), 6, 1867, 368-9. 
23M6m. Soc. Sc. Liege, (2), 3, 1873, 85-96. 
