456 HlSTOKY OF THE THEORY OP NUMBERS. [CHAP, XX
of m, but P is not divisible by b if nh is not a multiple of m. If b divides 10n— 1, P is divisible by b when /& = &, but not divisible by b when h is not a multiple of b.
A. Morel14 proved that the numbers ending with 12, 38, 62 or 88 are the only ones whose squares end with two equal digits.
H. Hoskins140 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. Plateau15 noted that every odd number not ending with 5 has a multiple of the form 11... 1 [Saint15].
P. Mansion16 proved the theorem of Plateau.
J. W. L. Glaisher17 deduced Crelle's7 theorem from Plateau's.16
G. A. Laisant18 treated a problem2 on reversing digits.
G. R. Perkins180 and A. Martin19 stated that all powers of numbers ending with 12890625 end with the same digits.
E. Catalan20 noted that the g. c. d. of two numbers of the form 1 ... 1 of n and n' digits is of like form and has A digits, where A is the g. c. d. of n and ri.
Lloyd Tanner, 20a generalizing Martin's19 question, found how many numbers N of n digits to the base r end with the same digits as their squares, i. e., N2—N=Krn. If rn is the product of q powers of primes, there are 2* -2 values of N. He20& found numbers M and N with n digits to the base r such that the numbers formed by prefixing M to N and N to M have a given ratio.
J. Plateau21 proposed the problem to find two numbers whose product has all its digits alike. Angenot noted that
bpq-l bp-l
&"-!' 6-1
give a solution for base 6. Catalan21 noted that Euler's theorem
for n prime to 6, furnishes a solution n} m.
Lloyd Tanner22 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.
14aMath. Quest. Educ. Times, 15, 1871, 89-91.
"Bull. Acad. Roy. de Belgique, (2), 16, 1863, 62; 28, 1874, 468-476.
16Nouv. 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.
"M&n. soc. sc. phys. et nat. de Bordeaux, (2), 1, 1876, 403-11. 18«Math. Miscellany, Flushing, N. Y., 2, 1839, 92. "Math. Quest. Educat. Times, 26, 1876, 28. 20M6m. Societ^ Sc. Liege, (2), 6, 1877, No. 4.
""Messenger Math., 7, 1877-8, 63^. Cases r£ 12, Math. Quest. Educ. Times, 28, 1878, 32-4 20bMath. Quest. Educ. Times, 29, 1878, 94-5. "Nouv. Corresp. Math., 4, 1878, 61-33. *Ibid., 5, 1879, 217; 6, 1880, 43.