CHAP. XX] PROPERTIES OP THE DIGITS OP NUMBERS. 461 as n — L If the cube of a number n of r digits ends with those r digits, 10r— n has the same property. P. Ziihlke46 proved the three theorems of Palmstrom39 and gave all solutions of xp^x (mod 103) for p = 3,. . .,12. M. Koppe47 noted that by prefixing a digit to a solution 0, 1, 5 or 6 of a?z=x (mod 10) we get solutions of x2=x (mod 102), then for 103, etc. We can pass from a solution with n digits for 10n to solutions with 2n digits for 102*. He treated also x5=x (mod 10W). G. Calvitti48 treated the problem: Given a number A, a set C of y digits, and a number p prime to the base g, to find the least number x of times the set C must be repeated at the right of A to give a number NX=A (mod p). The condition is G(Ni—N0)z=Q (mod p), where If iVi — -ZV0=0, any £ is a solution. If not, the least value X of x makes (r=0 (mod p/p), where p is the g. c. d. of NI—NQ and p. Then X is the 1. c. m. of Xi, . . . , Xft, where Xt is the least root of GssO (mod p*), if p/p is the product of pi, . . . , Pk, relatively prime in pairs. Hence the problem reduces to the case of a power of a prune p. Write (a)x for (a*— l)/(a— 1). It is shown that the least root of (a)ss=0 (mod pk) is mp*~', where m is the least root of (a)zssO (mod p), and p* is the highest power of p dividing (a)m. Given any set C of digits and any number p prime to the base g, there exist an infinitude of numbers C . . . C divisible by p. A. Ge'rardin480 added 220 to the sum of its digits, repeated the operation 18 times and obtained 418; 9 such operations on 284 gave 418. A. Boutin stated that if a and fc lead finally to the same number, neither, a nor b is divisible by 3, or both are divisible by 3 and not by 9, or both are divisible by 9. E. Malo49 considered periodicity properties of A and a in and solved Cesar o's26 three problems on the digits of powers of 5. A. L. Andreini50 noted that the squares of A and B end with the same p digits if and only if the smaller of r+s and u+v equals p, where «Sitz. Berlin Math. GeselL, 4, 1905, 10-11 (Suppl., Archiv Math. Phys., (3), 8, 1905). "Ibid., 5, 1906, 74-8. (Suppl., Archiv, (3), 11, 1907.) "Periodico di Mat., 21, 1906, 130-142. 48aSphinx-Oedipe, 1, 1906, 19, 47-8. Cf. i'intenn<$d. math., 22, 1915, 134, 215. "Sur certaines propri6t6s arith. du tableau des puissances de 5, Sphinx-Oedipe, 1906-7, 97-107; reprinted, Nancy, 1907, 13 pp., and in Nouv. Ann. Math., (4), 7, 1907, 419-431. "II Pitagora, Palermo, 14, 1907-8, 39-^7.