CONGRUENCE 155

This follows immediately from § 381, because 7 is a factor
of 103 4+ 1 and

10 = 3 (mod 7)
9 = 2 (mod 7)

r
whence r?

l
l

This rule may be put in a slightly different form, but one
apparently not so simple in application, as follows (1) :

GE( 2, S 2 a+2 2 Gk)

k=0 (mod 6) k=1 (mod 6) k=2 (mod 6)
= ( = ar + 3 % ar + 2 Z ak)
k=3 (mod 6) k=4 (mod 6) k=5 (mod 6)

(1) The rule in this form, when the result is zero, is equivalent to that
given in 1856 by Henri Monteux, ‘‘le célébre patre calculateur de la Tou-
raine,” ‘‘the celebrated shepherd computer of Touraine,” as quoted by
de Lapparent, Caractéres de Divisibilité des Nombres entiers, Mémoires
de la Société Impériale des Sciences Naturelles de Cherbourg, Vol. 1V,
1856, p. 247: ‘“Pour s’assurer si un nombre est divisible par 7, il faut le
séparer par tranches de trois chiffres, comme pour le lire. Puis, com-
mencant par la droit, multiplier le premier chiffre par 1, le second par 3
et le troisitme par 2. Agir de méme pour toutes les autres tranches,
jusqu’a la derniére. Additionner, ensuite, et séparément, les produits des
tranches paires et ceux des tranches impaires; enfin, faire la différence des
deux sommes. Si cette différence est nulle, le nombre est divisible par 7.”

Lapparent’s article is referd to by E. B. Elliott, On the Divisibility of
Numbers, the Mathematical Monthly, Vol. I, no. II, Nov. 1858. In this
article, p. 48, Elliott says: ‘“The above general method for finding the
numerical values of the remainders (7o, 71, 72, 73, + + +) peculiar to any given
divisor, accompanied by their numerical values for all integral divisors
between 1 and 30, was given by the present writer in the year 1846 to
classes of pupils at Lyons, in the State of New York.”

For these references I am indebted to Mr. L. L. Locke, of the Maxwell
Training College for Teachers, Brooklyn, N. Y.

It may be worth while to remark that I discoverd these rules independ-
ently, some years ago, by the method given above.

The reader will, I hope, pardon the length of this historical note, which
seems to be out of all proportion with its importance. My excuse is that
many persons seem to be fascinated by rules for divisibility by 7.