Chap. XX] PROPERTIES OF THE DiGITS OF NUMBERS. 463 
D. Biddle®^ applied congruences to find numbers like 15 and 93 whose 
product 1395 has the same digits as the factors. 
P. Cattaneo^^ considered numbers Q (and C) whose square (cube) ends 
with the same digits as the number itself. No Q>\ ends with 1. No two 
Q's with the same number of digits end with 5 or with 6. All Q's < 10^* are 
found. A single C of n digits ends with 4 or 6. Any Q is a C Any Q — 1 
is a C. If A?" is a Q with n digits and if 2N — 1 has n digits, it is a C. 
M. Thie,^^" using all nine digits >0, found numbers of 2, 3 or 4 digits 
with properties like 12483 = 5796. 
Pairs^^*^ of cubes 3^ 6^ and 375^ 387^ whose sums of digits are squares, 
32 and 61 
T. C. Lewis^^ discussed changes in the digits of a number to base r not 
affecting its divisibiUty by p. 
Numbers^'* B and 5" having the same sum of digits. 
Pairs^^ of primes like 23-89 = 29-83. 
Cases<^« like 7-9403 = 65821 and 3-1458 = 6-0729, where the digits 0, 
1,. . ., 9 occur without repetition. 
jy-pn+i ending" with the same digits as A^. 
Numbers^^ like 512 = (5+l+2)^ 47045881000000 = (47+4+58+81)^ 
AlP numbers like 2-5-27 = M8-15, 2+5+27 = 1 + 15 + 18. 
Number^° divisible by the same number reversed. 
Number^^ an exact power of the sum of its digits; two numbers each 
an exact power of the sum of the digits of the other. 
Solve'^^ KN-{-P = N', N' derived from N by reversing the digits. 
Symmetrical numbers (ibid., p. 195). 
F. Stasi^^ proved that, if a, h are given integers and a has m digits, we 
can find a multiple of b of the form 
10''(a-10'"'+a-10"^^'-^^+ . . . +a), p^O. 
Taking b prime to a and to 10, we see that b divides 10""'+ ... +1. The 
case m= 1 gives the result of Plateau. ^^ 
Cunningham'^^" and others wrote iVi for the sum of A^ and its digits to 
base r, iV2 for the sum of A^i and its digits, etc., and found when N^ is 
divisible by r — 1 . • 
"Math. Quest. Educ. Times, (2), 19, 1911, 60-2. Cf. (2), 17, 1900, 44. 
«2Periodico di Mat., 26, 1911, 203-7. 
fi^^Nouv. Ann. Math., (4), 11, 1911, 46. 
62''Sphinx-Oedipe, 6, 1911, 62. 
"Messenger Math., 41, 1911-12, 185-192. 
6*L'intermMiaire des math., 18, 1911, 90-91; 19, 1912, 267-8. 
^Ubid., 1911, 121, 239. ««/6id., 19, 1912, 26-7, 187. 
"Ibid., 50-1, 274-9. 
^^Ibid., 77-8, 97. 
«9/6id., 125, 211. 
''"Ibid., 128. 
''Ibid., 137-9, 202; 20, 1913, 80-81. 
"76id., 221. 
"II Boll. Matematica Gior. Sc.-Didat., 11, 1912, 233-5. 
73«Math. Quest. Educ. Times, (2), 21, 1912, 52-3. 
