CHAPTER XX. 
PROPERTIES OF THE DIGITS OF NUMBERS. 
John HilP noted that 139854276=118262 is formed of the nine digits 
permuted and believed erroneously that it is the only such square. 
N. BrownelP" found 169 and 961 as the squares whose three digits are 
in reverse order and whose roots are composed of the same digits in reverse 
order. The least digit in the roots is also the least in the squares, while the 
greatest digit in the roots is one-third of the greatest in the squares and 
one-half of the digit in the tens place. 
W. Saint^'' proved that every odd number N not divisible by 5 is a divisor 
of a number 11 . . .1 oi D^N digits [by a proof holding only for N prime 
also to 3]. For, let 1 ... 1 (to D digits) have the quotient q and remainder 
r when divided by D. This remainder r must recur if the number of digits 
1 be increased sufficiently. Hence let 1...1 (to D+d digits) give the 
remainder r and quotient Q when divided by D. By subtraction, D{Q — q) 
= 1. . .10. . .0 (with d units followed by D zeros). Hence if 1. . .1 (to d 
digits) were not divisible by every odd number ^ D and prime to 5 [and to 
3], there would be a remainder R; then RO. . .0 (with D zeros) would be 
divisible by an odd number prime to 5 [and to 3], which is impossible. 
P. Barlow^" stated, and several gave inadequate proofs, that no square 
has all its digits alike. He^'^ stated and proved that 111111111^=1 23456- 
78987654321 is the largest square such that if unity be subtracted from each 
of its digits and again from each digit of the remainder, etc., all zeros being 
suppressed, each remainder is a square. Denote (10^ — 1)/(10 — 1) by [k]. 
Then |^(x+l)p has x digits and exceeds [x] by 10{|(a: — l)p. Since 
zeros are suppressed we have a square as remainder, and the process can 
be repeated. It is stated that therefore the property holds only for V, 
IV, IIP, .... 
Several^* found that 135 is the only number N composed of three digits 
in arithmetical progression such that the digits will be reversed if 132 times 
the middle digit be added to N. 
W. Saint ^•'^ found the least integral square ending with the greatest num- 
ber of equal digits. The possible final digits are 1, 4, 5, 6, 9. Any square is 
of the form 4n or 4n+l. Hence the final digit is 4. If the square termi- 
nated with more than three 4's, its quotient by 4 would be a square ending 
with two I's, just proved to be impossible. Of the numbers ending with 
^Arithmetic, both ia Theory and Practice, ed. 4, London, 1727, 322. 
i«The Gentleman's Diary, or Math. Repository, London, 1767; Davis' ed., 2, 1814, 123. 
ifcJour. Nat. Phil. Chem. Arts (ed., Nicholson), London, 24, 1809, 124-6. 
I'^The Gentleman's Diary, or Math. Repository, London, 1810, 38-9, Quest. 952. 
^^Ibid., 1810, 39-40, Quset. 953. 
^^Ibid., 1811, 33-4, Quest. 960. 
I/Ladies' Diary, 1810-11, Quest., 1218; Leybourn's M. Quest. L. D., 4, 1817, 139^1. 
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