Chap. XX] PROPERTIES OF THE DiGITS OF NXJMBERS. 455 
quotient is 46|. 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 synunetrical number abcba of five digits whose square exhibits 
all ten digits, W. Rutherford^" 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-{-b, c=S,he got 35853. 
J. Sampson noted also the answers 84648, 97779, 
J. A. Grunert^ proved by use of Euler's generahzation of Fermat's the- 
orem that^ every number divides 9 . . . 90 . . .0. ---^ 
Drot^" asked for the values of x for which N"" has the same final k digits 
as N, when ^ = 1, 2 or 3. 
J. Bertrand^'' discussed the numbers of digits of certain numbers. 
A. G. Emsmann^ treated a number 6 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 b the remainder shall equal the number obtained 
by writing the digits of b in reverse order. 
J. Booth^° 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. Bianchi^^" noted various numerical relations like 10^ = 11111111 + 
8.1111111 + 8.9.111111+... + 8.9M + 9^ = 2222222 + ...+7.8^2+8^ 98 = 
(12-1-0)9-1, 987 = (123-12-l)9-3, 9876 = (1234-123-13)9-6. 
C. M. Ingleby^^ 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, of a single digit; and proved that S{MN)=S{SM-SN). 
P. W. Flood^^" proved that 64 is the only square the sum of whose digits 
less unity and product plus unity are squares. 
G. Cantor^^ employed any distinct positive integers a, 6, ... , considered 
the system of integers in which a occurs a times, b occurs b 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,. . ., a; /3 = 0, 1, . . ., 6; . . .. 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. Barillari^^ noted that, if 10 belongs to the exponent m modulo b, 
the number P = a/3 . . . Xa/3 ... X ... , obtained by repeating h times (/i > 1) any 
set of n digits, is divisible by 6 if 6 is prime to 10^-1 and if nh is a multiple 
'^Ladies' Diary, 1835, 38, Quest. 1576. 
8Jour. fur Math., 5, 1830, 185-6. 
^'^Nouv. Ann. Math., 4 1845, 637-44; 5, 1846, 25. For references to tables of powers, 13, 
1854, 424-5. 
oblbid., 8, 1849, 354. 
"Abhandlung liber eine Aufgabe aus der Zahlentheorie, Progr. Frankfurt, 1850, 36 pp. 
loProc. Roy. Soc. London, 7, 1854-5, 42-3. 
^""Proprieta e rapporti de' numeri interi e composti coUe 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 DubUn Messenger Math., 3, 1866, 30-31. 
"''Math. Quest. Educ. Times, 7, 1867, 30. 
"Zeitschrift Math. Phys., 14, 1869, 121-8. 
"Giomale di Mat., 9, 1871, 125-135. 
