of it will be 1; for, if this feries be multiplied by the divisor r-1, it will produce the dividend "-1. It will appear alfo by performing the division, and inserting for n any number.

r

LEM. 2. Let be any number, and n any integer odd number, +1 is divifible by r+1. Alfo, if n is any even number, -1 is divisible by r+1.

The quotient in both cafes is "gribam 24 +3, &c. till the exponent of r be o, and the laft term 1. If this feries confift of an odd number of terms, and be multiplied by r+1 the divifor, the product is "I the dividend. If the feries confift of an even number of terms, the product is "-1; but it is plain that the number of terms will be odd only when ʼn is odd, and even only when n is even. The conclufion will be manifeft by performing the divifion.

LEM. 3. 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, a+br +cr2+dr3+era, &c.

THEOR. IV. If from any number in the general scale now described, the fum of its digits be fubtracted, the remainder is divifible by r-I.

The number is a+br+cr2+dr3, &c. and the fum of the digits is a+b+c+d, &c. Subtracting the latter from the former, the remainder is br―b+cr2-c+dr3 ―d,&c.=bxr-1+cxr2—1+dxr3—I &c. But (by Lem. 1.) "—1 is divisible by r-1, whatever integer number n may be, and therefore any multiple of "-1 is also divifible by r—I. Hence each of the terms, bxr-1, cXI, &c. is divisible by r-1: and therefore the whole is divifible by r-I.

COR. I. Any number, the fum of whose digits is divifible by r—1, is itself divisible by r-1. Let the number be called N, and the fum of the digits D; then, by this Prop. N-D is divisible by r—1, and D is supposed

posed to be divisible by r-1; therefore it is plain that N must also be divifible by

I.

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

COR. 3. This theorem, with the corollaries, relates to any fcale whatever. It includes therefore the well known property of 9, and of 3 its aliquot part, in the decimal fcale; for, fince r=10, r1=9.

THEOR. V. In any number, if from the fum of the coefficients of the odd powers of r, the fum of the coefficients of the even powers be fubtracted, and the remainder added to the number itself, the fum will be divifible by r+1.

In the number a+br+cr2+dr3+er+† fr3, &c. the fum of the coefficients of the odd powers of r is b+d+f, &c. the sum of the coefficients of the even powers of r is a+c+e, &c. If the latter fum be fubtracted from the former, and the remainder added to the given number, it makes br+b+ cr2—c+dr3+d+ér*—e+fr$+f,&c.=bx r+i+cxr2=1+dxr3+i+exra—1+ fxr+1,&c. But (by Lem 2.)r+1,r2—1, r3+1, &c. are cach divifible by r+1, and therefore any multiples of them are also divisible by r+1, hence the whole number is divifible by r+1.

Cor. I.

If the difference of the fum of the even digits, and the fum of the odd digits of any number be divisible by r+1, the number itself is divisible by r+1.

Let the fum of the even digits, (that is, the coefficients of the odd powers of r) be D, the fum of the odd digits be d, and let the number be N. Then, by the theorem, N+D-d is divifible by r+1, and it is fuppofed that D-d is divifible by r+1; [therefore N is divifible by r+1.

Cor.

Cor. 2. In like manner, if D-d is divifible by an aliquot part of r+1, 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 powers of r, and if the fum of its digits be divisible by r+1, that number is divisible by r+1.

Cor. 4. In the common fcale r+1=11, 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 11, the given number therefore (Cor. 1.) is divisible by 11. Thus alfo, the fum of the digits of 7040308 is divisible by 11, and therefore the number is divifible by 11. (Cor. 3.).

SCHOLIU M.

These theorems relate to any scale whatever, and therefore the properties of r-1