456 History of the Theory of Numbers. [Chap. XX 
of w, but P is not divisible by h if nh is not a multiple of m. If h divides 
10"— 1, P is divisible by b when h = b, but not divisible by b when h is not a 
multiple of b. 
A. MoreP* proved that the numbers ending with 12, 38, 62 or 88 are the 
only ones whose squares end with two equal digits. 
H. Hoskins^^'* found the sum of the 117852 numbers of 7 digits which 
can be formed with the digits 1, 1, 2, 2, 2, 2, 2, 3, 3, 4, 5, 6, 7. 
J. Plateau^^ noted that every odd number not ending with 5 has a multiple 
of the form 11... 1 [Saint^^. 
P. Mansion^^ proved the theorem of Plateau. 
J. W. L. Glaisher^^ deduced Crelle's^ theorem from Plateau's.^^ 
C. A. Laisant^^ treated a problem^ on reversing digits. 
G. R. Perkins^^" and A. Martin^^ stated that all powers of numbers end- 
ing with 12890625 end with the same digits. 
E. Catalan^" noted that the g. c. d. of two numbers of the form 1 ... 1 
of n and n' digits is of Uke form and has A digits, where A is the g. c. d. of n 
and n'. 
Lloyd Tanner,^"" generalizing Martin's^^ question, found how many 
numbers N oin digits to the base r end with the same digits as their squares, 
i. e., N^—N = Kr''. If r" is the product of q powers of primes, there are 
2^ — 2 values of N. He^^^ found numbers M and N with n digits to the 
base r such that the numbers formed by prefixing M to N and iV to M 
have a given ratio. 
J. Plateau^ ^ proposed the problem to find two numbers whose product 
has all its digits alike. Angenot noted that 
6^-1' 6-1 
give a solution for base b. Catalan^^ noted that Euler's theorem 
6-1 
nm 
for n prime to 6, furnishes a solution n, m. 
Lloyd Tanner" stated and Laisant proved that 87109376 and 12890625 
are the only numbers of 8 digits whose squares end with the same 8 digits. 
"Nouv. Ann. Math., (2), 10, 1871, 44-6, 187-8. 
i^Math. Quest. Educ. Times, 15, 1871, 89-91. 
"BuU. Acad. Roy. de Belgique, (2), 16, 1863, 62; 28, 1874, 468-476. 
"Nouv. Corresp. Math., 1, 1874-5, 8-12; Mathesis, 3, 1883, 196-7. Bull. Bibl. Storia Sc. 
Mat., 10, 1877, 476-7. 
"Messenger Math., 5, 1875-6, 3-5. 
**M6m. 80C. sc. phys. et nat. de Bordeaux, (2), 1, 1876, 403-11. 
i«»Math. Miscellany, Flushing, N. Y., 2, 1839, 92. 
"Math. Quest. Educat. Times, 26, 1876, 28. 
»oM6m. Society Sc. Li^ge, (2), 6, 1877, No. 4. 
"•^Messenger Math., 7, 1877-8, 63-4. Cases r^ 12, Math. Quest. Educ. Times, 28, 1878, 32-4. 
Jo'-Math. Quest. Educ. Times, 29, 1878, 94-5. 
«Nouv. Corresp. Math., 4, 1878, 61-63. 
«/Wd., 5, 1879, 217; 6, 1880, 43. 
