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of it will be 1; for, if this series be multi-
plied by the divisor r-1, it will produce
the dividend pop-1. It will appear also by
performing the division, and inserting for n

.

any number

3

1

Lem. 2. Letr be any number, and n any integer odd number, poti ti is divisible by rti. Also, if n is any even number, gun.__-is divisible by rti.

The quotient in both cases is gon' gutham2 十re ?, &c. till the exponent of r be o, and the last term goIIf this series con-. fift of an odd number of terms, and be multiplied by rti the divisor, the product is gun to I the dividend. If the series conGift of an even number of terms, the pro- . duct is ghal; but it is plain that the number of terms will be odd only when n is odd, and even only when n is even. The conclusion will be manifest by performing the division.

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If r is the root of an arithmetical scale, any number in that scale

may

be represented in the following manner, a, b, c,

&c.

&c. being the coefficients or digits, atbr tcra+driterè, &c.

THEOR. IV. If from any number in the general scale now described, the sum of its digits be subtracted, the remainder is divisible by ri.

The number is atbr+cra+dr?, &c. and the sum of the digits is atb+c+d, &c. Subtracting the latter from the former, the remainder is bræbtract dr3 _d, &c.=bxrritoxp? - 1+dx.m3—1 &c. But (by Lem. 1.) yn — 1 is divisible by powi, whatever integer number n may be, and therefore any multiple of grami is also divisible by r_l.

Hence each of the terms, bxr-I, CX72—1, &c. is divisible by r-I: and therefore the whole is divifible by r-ri.

1

CoR. I. Any number, the sum of whose digits is divisible by r-1, is itself divisible by r-1. Let the number be called N, and the sum of the digits D; then, by this Prop. N-D is divisible by r-1, and D is sup

posed

posed to be divisible by r-I; therefore it
is plain that N must also be divisible by

I.

COR. 2. Any number, the sum of whose digits is divisible by an aliquot part of r-1, is also divisible by thai aliquot part. For, let N and D denote as before; and since N-D (Theor. 4.) is divisible by r-1, it is also 'divisible by an aliquot part of r-1; but D is divisible by an aliquot part of r-1 therefore' N is also divisible by that aliquot part.

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3.

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Cor. This theorem, with the corollaries, relates to any scale whatever. It includes therefore the well known property of 9, and of 3 its aliquot part, in the decimal scale; for, since r=10,-I=9.

powers of

THEOR. V. In any number, if from the
sum of the coefficients of the odd
1,

the sum of the coefficients of the even
powers be subtracted, and the remainder
added to the number itself, the sum will be
divisible by r+i.

Ia

In the number a +br +cr? +drš toert frs, &c. the sum of the coefficients of the odd powers of r is 6+d+f, &c. the sum of the coefficients of the even powers of r is atote, &c. If the latter fum be fubtracted from the foriper, and the remainder added to the given number, it makes br+b+ cramc+dri+d+értmetfps +f,&c.=bx stitcXperi+d XmFitexpt-it fxr+1, &c. But (by Lem 2.)r +1,92-1, godt 1, &c. are each divisible by rti, and therefore any multiples of them are also divisible by rti, hence the whole number is divisible by rti.

Cor. I. If the difference of the sum of the even digits, and the sum of the odd di

any

number be divisible by rti, the number itself is divisible by rti.

Let the sum of the even digits, (that is, the coefficients of the odd powers of r) be D, the sum of the odd digits be d, and let the number be N. Then, by the theorem, N+D-d is divisible by rt1, and it is supposed that D-d is divisible by rti; therefore N is divisible by rti.

Cor.

gits of

powers of

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Cor. 2. In like manner, if D-d is divi-
fible by an aliquot part of rti, N will be
divisible by that aliquot part.
Cor.
3.

If a number want all the odd
powers of r, or if it want all the even

r,

and if the sum of its digits be
divisible by rtı, that number is divisible
by rti.
Cor.

4.

In the common scale r+I=II, which therefore will have the properties mentioned in this theorem, and the corollaries. Thus, in the number 64834, the sum of the even digits is 7, the sum of the odd digits is 18, and the difference is 11, a number divisible by II, the given number therefore (Cor. 1.) is divisible by 11. Thus also, the sum of the digits of 7040308 is divisible by it, and therefore the number is divisible by 11. (Cor. 3.).

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SCHOLIU M.

These theorems relate to any scale whatever, and therefore the properties of r-I T

in

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