CHAPTER XII. 
CRITERIA FOR DIVISIBILITY BY A GIVEN NUMBER. 
In the Talmud^ lOOa+6 is stated to be divisible by 7 if 2a+b is divis- 
ible by 7. 
Hippolytos^", in the third century, examined the remainder on the 
division of certain sums of digits by 7 or 9, but made no appHcation to 
checking numerical computation. 
Avicenna or Ibn Sina (980-1037) is said to have been the discoverer 
of the familiar rule for casting out of nines (cf . Fontes^^) ; but it seems to 
have been of Indian origin.-^'' 
Alkarkhi^^ (about 1015) tested by 9 and 11. 
Ibn Musa Alchwarizmi^'* (first quarter of the ninth century) tested by 9. 
Leonardo Pisano^^ gave in his Liber Abbaci, 1202, a proof of the' test 
for 9, and indicated tests for 7, 11. 
Ibn Albanna^-'^ (born about 1252), an Arab, gave tests for 7, 8, 9. 
In the fifteenth century, the Arab Sibt el-Maridini^'' tested addition by 
casting out multiples of 7 or 8. 
Nicolas Chuquet^^ in 1484 checked the four operations by casting out 9's. 
J. Widmann^'' tested by 7 and 9. 
Luca Paciuolo^ tested by 7, as well as by 9, the fundamental operations, 
but gave no rule to calculate rapidly the remainder on division by 7. 
Petrus Apianus^" tested by 6, 7, 8, 9. 
Robert Recorde^'' tested by 9. 
Pierre ForcadeP noted that to test by 7 = 10 — 3 we multiply the first 
digit by 3, subtract multiples of 7, add the residue to the next digit, then 
multiply the sum by 3, etc. 
Blaise Pascal^ stated and proved a criterion for the divisibility of any 
number N by any number A. Let ri, r2, 7*3, . . . , be the remainders obtained 
when 10, lOfi, lOrg, ... are divided by A. Then iV = a+ 106 + 100c+ ... is 
divisible by A if and only if a-\-rib-\-r2C+ . . . is divisible by A. 
'Babylonian Talmud, Wilna edition by Romm, Book Aboda Sara, p. 96. 
i«M. Cantor, Geschichte der Math., ed. 3, I, 1907, 461. 
^^Ibid., 511, 611, 756-7, 763-6. 
i^Cf. Carra de Vaux, Bibliotheca Math., (2), 13, 1899, 33-4. 
i<^M. Cantor, Geschichte der Math., ed. 3, I, 1907, 717. 
i^Scritti, 1, 1857, 8, 20, 39, 45; Cantor, Geschichte, 2, 1892, 8-10. 
'/Le TaUfhys d'Ibn Albanna public et traduit par A. Marre, Atti Accad. Pont. Nuovi Lincei, 
17, 1863-4, 297. Cf. M. Cantor, Geschichte Math., I, ed. 2, 757, 759; ed. 3, 805-8. 
iffLe Triparty en la science de nombres. Bull. Bibl. St. Sc. Math., 13, 1880, 602-3. 
^''Behede vnd hubsche Rechnung . . . , Leipzig, 1489. 
^Summa de arithmetica geometria proportion! et proportionalita, Venice, 1494, f. 22, r. 
2"Ein newe. . .Kauffmans Rechnung, Ingolstadt, 1527, etc. 
^^The Grovnd of Artes, London, c. 1542, etc. 
^L'Arithmeticqve de P. Forcadel de Beziers, Paris, 1556, 59-60. 
*De numeris multiphcibus, presented to the Acad^mie Parisienne, in 1654, first published in 
1665; Oeuvres de Pascal, 3, Paris, 1908, 311-339; 5, 1779, 123-134. 
337 
