CHAP. XX] PROPERTIES OF THE DIGITS or NUMBERS. 455 quotient is 46J. By the last condition, the middle digit must be 3, since not a higher multiple of 3. Hence the number is 931. To find a symmetrical number dbcba of five digits whose square exhibits all ten digits, W. Rutherford7' noted that the square is divisible by 9 since the sum of the digits is divisible by 9. Hence the sum of the digits of the number is divisible by 3. Also a ^ 3. Taking c =a+6, c =8, he got 35853. J. Sampson noted also the answers 84648, 97779. J. A. Grunert8 proved by use of Euler's generalization of Fermat's theorem that6 every number divides 9.. .90.. .0. Drot8a asked for the values of x for which Nx has the same final k digits as N, when k — 1, 2 or 3. J. Bertrand8* discussed the numbers of digits of certain numbers. A. G. Emsmann9 treated a number b of n digits to base 10 equal to the product of the sum of its digits by a, and such that if another number of n digits be subtracted from 6 the remainder shall equal the number obtained by writing the digits of 6 in reverse order. J. Booth10 noted that a number of six digits formed by repeating any set of three digits is divisible by 7, 11, 13 [since by 1001]. G. Bianchi10" noted various numerical relations like 109 = 11111,111+ 8.1111111 + 8.9.111111+... + 8.96.l + 98=?22?g22J-f...+7.86.2+88, 98=* (12-1-0)9-1, 987 = (123-12-1)9-3, 9876 = (1234-123-13)9-6. C. M. Ingleby11 added the digits of a number N written to base r, then added the digits of this sum, etc., finally obtaining a number, designated SN,'oi a single digit; and proved that S(MN)=S(SM-SN). P. W. Flood110 proved that 64 is the only square the sum of whose digits less unity and product plus unity are squares. G. Cantor12 employed any distinct positive integers a, 6,..., considered the system of integers in which a occurs a times, b occurs 5 times, etc., and called a system simple if every number can be expressed in a single way in the form aa+/36+. .., where a = 0, 1,..., 5; 0 = 0, !,...,£;---- A system is simple if and only if each basal number k divides the next one I and if k occurs k = (l/k) — 1 times. G. Barillari13 noted that, if 10 belongs to the exponent m modulo 6, the number P = aft...Xa/3...X..., obtained by repeating h times (h>l) any set of n digits, is divisible by b if 6 is prime to 10n—1 and if rih is a multiple ^Ladies' Diary, 1835, 38, Quest. 1576. "Jour, fur Math., 5, 1830, 185-6. ^Nouv. Ana. Math., 4 1845, 637-44; 5, 1846, 25. For references to tables of powers, 13, 1854, 424-5. •ft/bid., 8, 1849, 354. 'Abhandlung iiber eine Aufgabe aua der Zahlentheorie, Progr. Frankfurt, 1850, 36 pp. 10Proc. Roy. Soc. London, 7, 1854-5, 42-3. 100Proprieta e rapporti de' numeri interi e compost! colle cifre semplici . . . , Modena, 1856. Same in Mem. di Mat. e di Fis. Soc. Ital. Sc., Modena, (2), 1,1862, 1-36, 207. "Oxford, Cambr. and Dublin Messenger Math., 3, 1866, 30-31. lwMath. Quest. Educ. Times, 7, 1867, 30. "Zeitschrift Math. Phys., 14, 1869, 121-8. "Giornale di Mat., 9, 1871, 125-135.