Annals of Mathematics

On the Divisibility or Non-Divisibility of Numbers by SevenAuthor(s): Alexander EvansSource: The Analyst, Vol. 10, No. 5 (Sep., 1883), pp. 134-135Published by: Annals of MathematicsStable URL: http://www.jstor.org/stable/2635785 .

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?134?

becomes infinite, any finite point may be regarded as at an infinitesimal dis? tance from the circumference relatively to the infinite radius.

15. There is a special case of the complex anharmonic ratio which is

worthy of notice, in which, as in the case of the harmonic ratio, there are less than six different values of the anharmonic ratio. In the harmonic ra?

tio, there are in fact three different values, viz., ?1, 2, and J, the first of

these numbers being its own reciprocal, the next being (in the nomenclature of my articles, Analyst Vol. IX, p. 185, and Vol. X, p. 76) its own con?

jugate, and the third its own complement. The only other case in which there are but three different values is that of 1, 0, and oo, which occurs

when two ofthe four points are coincident. In the case in question how?

ever there are but two different values, namely J ? J^j/3, each of which is

at once the reciprocal, complement and conjugate of the other. If the fun?

damental points A9 B and C are given, two points P and Pf may be found

such that P A A Pf A BC

and BC

shall have these two values, and then each of the anharmonic ratios of the

points ABCP or ABCP' will have one ofthe two values, so that the inter?

change of any two points of the four has the same effect. Since for these val?

ues, which are e?liK9 R ? 1 (see ? 10) /> = p0 ; hence, denoting the sides of

the triangle by a, b, c and the distances from P to the vertices A, B, C by

af, bf and cf, we have aaf ? bbf = ccf; and, since 6= ?^?r, the difference

between the angles BCA and BPA (reckoned in the same direction), QAP and QBP, etc, is in each case equal to 60?.

If ABC is an isosceles triangle, P and Pf are on the bisector of the angle

opposite the base. If one of the angles of the triangle is 60? or 120?, one

of the points is on the side opposite this angle; but if the triangle is equi? lateral this point is at infinity, the other being the centre ofthe triangle.

ON THE D1VISIBIL1TY OR N0N-DIVISIB1LITY

OF NUMBEBS BY SEVEN.

BY ALEXANDER EVANS, ESQ., ELKTON, MARYLAND.

The test for divisibility by seven has been considered in relation to the

number of digits composing the dividend: As "for two or three figures; for three or four figures; for five figures; for five, six, seven or more figures."

A writer, after so treating the subject, says, "the various tokens of seven

are not so dimcult to remember as at first may appear".

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?135?

In the examples of the processes of Arithmetic and Algebra, by the Soci-

ety for the Diffusion of Useful Knowledge, the author, at page two, says, division by seven is "according to no rule sufficiently simple to be useful."

The writer of this article submits the following rule:

Multiply the right hand, or units digit, by nine, and deduct this product from the remaining digits, treat the remainder in the same way, multiply? ing its units digit by nine and deducting the product from the remaining digits, until a number is arrived at which is evidently divisible or not divisi? ble by seven; according as the one or the other occurs the original number

is, or is not divisible by seven.

Examples.

1. 2121 9 =1X9

203 27 = 3X9 7

Hence 2121 is divisible by 7.

13433 27 =3X9

1316 54 =6X9 77

Hence 13433 is divisible by 7. This process, though only given as a rule, and not for its practical use?

ful ness, is often easier when "working in the head" than the direct method. 3. 99009

81 =9X9 9819 81 =9X9

900 9 Hence 99009 is not divisible by 7.

If 12 be used as a multiplier instead of 9, then the above rule becomes a test for divisibility by 11. Also, 22 is the multiplier and test for divisibilisy by 13; and 39 is the multiplier and test for divisibility by 17.

It will be observed that the least test multiplier has not always been used; this is because the least does not always lead quickest to the result.

For divisibility by every prime number, test multipliers may be found. A list extending as far as 43 is given below.

For divisibility by 7,

For an art on this subject by Prof. Brooks, see Analyst, Vol. II, p. 129.

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