Chap. XII] CRITERIA FOR DIVISIBILITY. 343 
To testiV = ao+Oi-S+ • • • +On5"for the divisor D prime to B, determine d and 
X so that Bd = Dx+l. Multiply this equation by Oq and subtract from N. 
Thus 
N=-BN'-DaoX, N' = aod-\-{ai+a2B-\- . . . +a^B''-^)B. 
Hence N is divisible by D if and only if N' is divisible by D. Now, N' is 
derived from N by supressing the units digit do and adding to the result the 
product aod. Next operate with N' as we did with A^. 
J. Malengreau^^ would test N for a factor q prime to 10 by seeking a 
multiple 11 ... 1 (to m digits) of q, then an exponent t such that the number 
of digits of lO'-A^ is a multiple of m. From each set of m digits of lO'-A^ 
subtract the nearest multiple of 1 ... 1 (to m digits) . The sum of the resi- 
dues is divisible by q if and only if A'' is divisible by q. 
G. Loria^^ proved that N = aQ-\-gai-\- . . .-\-g''ak is divisible by a if and 
only if a divides the sum ao+ • • . +a^ of the digits of N written to a base g of 
the form /ca+ 1 ; or if a divides Oq — ai +02 — • • • when the base g is of the form 
ka — 1. Taking g = IC", we have the test, in Gelin's Arithm^tique, in terms 
of groups of m digits. We may select m to be |0(a) or a number such that 
lO'^il has the factor a. Inplace of 00+^1+ • • . wheng' = 10'", we may employ 
pao+Xai + 10Xa2+ . . . +10"-'Xa^_i 
+ s\o^™-^(a,^+10a,^+i+ . . . +10"'-'a,^+m-i), 
k = l 
where X = l, 2 or 5, and p is determined by 10p/X=l (mod o). Taking 
a = 7, 13, 17, 19, 23, special tests for divisors are obtained. 
G. Loria^^ proved that, if ao, ai,. . . are successive sets of t digits of N, 
counted from the right, and o- = ao='=cti+02=^«3+ • • •, then 
N-(T = a,{10'=Fl)-\-a2{10^'-l)+as{10^'=pl) + . . ., 
so that a factor of 10'=f1 divides A^ if and only if it divides a. 
A. Tagiuri^^ extended the last result to any base g. We have 
N = ao+ga,+ . . . =Nom+g"'Nr^+g''^N2^-\- . . . 
if N,m = o,pm+apm+i9-\- • • • +«pm+m-i^"'"^ Heuce, if 9"^= ± 1 (mod a), 
N=Nom^Nim+N2m=^... (mod a). 
L. Ripert^" noted that lOD-\-uis divisible by lOS+i if Di—bu is divisible, 
and gave many tests for small divisors. 
G. Biase^^ derived tests that \Od-\-u has the factor 7 or 19 from 
2{l{)d+u)^2u-d (mod 7), 2{lQd+u)=2u+d (mod 19). 
O. Meissner^^ reported on certain tests cited above. 
"Mathesis, (3), 1, 1901, 197-8. 
*^Rendiconti Accad. Lincei (Math.), (5), 10, 1901, sem. 2, 150-8. Mathesis, (3), 2, 1902, 33-39. 
"II Boll. Matematica Gior. Sc.-Didat., Bologna, 1, 1902. Cf. A. Bindoni, ibid., 4, 1905, 87. 
"Periodico di Mat., 18, 1903, 43-45. ^oL'enseignement math., 6, 1904, 40-46. 
"II Boll. Matematica Gior. Sc.-Didat., Bologna, 4, 1905, 92-6. 
^''Math. Naturw. Blatter, 3, 1906, 97-99. 
