July 11, 1884.] 



• KNOWLEDGE • 



39 



TESTS OF DIVISIBILITY. 



(A reply to the general question involved in Letter 1271.) 



[1334] — To determine whether a number (N) ia divisil)le without 

 remainder bv another number n (n being any number prime 

 to 10). 



1. Divide 009 .... by n until there ia no remainder. It ean be 

 algebraically proved that this will happen when (« — 1) S)'s have 

 been iised. In practice a smaller number will often be found 

 sufficient. For instance, if 7!. = 41, only five 9's are needed, 



2. Count the number of O'a used in the division, and mark off K 

 in periods of the same number of figures, commencing from the 

 right. 



3. Add the periods together, marking off and adding again if the 

 immber of figures in the sum exceed the proper number in a 

 period. 



3a. When the period consists of an even number of figures, mark 

 off the sum just obtained into semi-periods, and subtract one from 

 the other. (This is in principle the same thing as the rule in 

 Letter 127-1, where alternate semi-periods are added in the first 

 place, but I think my method will be found a little shorter in 

 pi-actice.) 



4. Divide the final result in 3 (or 3a) by n. If there be no 

 remainder, N is divisible by n. W. 



PROPERTY OF NUMBERS. 



[1335] — Thank you for publishing the problem under the above 

 title (1295, p. 424), which if not dithcult is novel and interesting 

 1 solve it as follows ; — Let the remainders upon division by 3, 5, 7, 

 and 11 be t, /, s, and e, respectively, then the number ia 3S5i + 231/ 

 + 33Uv + 210e, less auch a multiple of 1155 as must bo subtracted to 

 reduce the number to one of three figures. It is evident that a 

 number expressed by the above formula will divide by each of the 

 <livisors with the given remainder, and that no other number will do 

 so unless the difference between it and this number is divisible by 

 3, 5, 7, and 11, that is, is a multiple of 1155. 



It is easy to construct formulas for similar problems with different 

 divisors. A limit must be fixed so that the number shall not 

 exceed the product of all the divisors. [And the divisors must 

 have no common factors.] Then the co-eliicient of each remainder 

 must be a number which divides by the divisor, producing that re- 

 mainder with 1 remainder, and by all the other divisors without 

 remainder. If the divisors were 7, 11, and 13 the co-efficients 

 would be 715, 304. and 924. 



The principle of this problem suggests to me a trick to bo played 

 with cards, as follows : — 



Mix together two whole packs of cards and request some one to 

 cut the double pack into two portions, and to divide one portion 

 into seven equal packets, adding the superfluous cards, if any, to 

 the other portion, then leaving the seven packets to divide the 

 other portion into five equal packets, handing any superfluous cards 

 to the performer of the trick, then mixing the five packets to 

 divide again into three equal packets, again handing any super- 

 fluous cards to the performer, who will thereupon state how many 

 cards are contained in each of the three packets and in each of the 

 seven packets. Perhaps some of your readers may like to search 

 for the solution of this trick. Algernon- Bray. 



New York, June 18, 18S4. 



AN ANIMAL ATTEMPTS SUICIDE. 



[1336] — Did not the poor animal referred to in letter 1308, 

 already suffering from heat, and smoke, and fright, mistake the 

 broken glass for ice or water, and carry it to its mouth with the 

 hope of relieviig its sufferings, rather than with the idea of 

 suicide ? M. J. C. 



COINCIDENCES, &c. 



[1337] — Most people have heard of the " War Office Ghost," the 

 story of which is given in " Footfalls on the Boundaries of Another 

 World," — how the wife of an officer who was killed at Sir Colin 

 Campbell's relief of Lucknow saw him at Cambridge the night of 

 his death, and thereby detected an error of one day in the date as 

 given in the official returns. Now, in the published account it is 

 mentioned that he appeared to her with his shirt-front stained with 

 blood, but it is most unusual for an officer in regimental tiniform to 

 show any shirt front at all. It happens, however, that shortly 

 before his death he had a fancy patrol coat made, the front of 

 which was cut open, showing a great deal of linen. I had been 

 with him at Cawnpore a week or two before he was killed, and 



particularly noticed thia. It would seem as if his wife actually 

 saw him in the very dress he was killed in, without having pre- 

 viously been aware of the peonliarity of that dress. By-the-bye, it 

 should have been stated at what hour of the day he was killed, so 

 as to allow for difference of longitude. Thus, if his death happened 

 at 8 a.m. his wife ought to have seen him between 1 and 2 a.m. 



MUSAFIB. 



[1338] — On Saturday, June 7, I had been writing to a young 

 lady, on her death-bed, as it proved. I had finished my letter a few 

 moments before one o'clock. Just at this moment she awoke from 

 a dosing sleep. She called her sister, and asked her if she had 

 been writing a letter to her. She replied she had not. " Then," 



replied the dear girl, " if you have not Mr. W has, and he has 



blotted it in two places." She died early on the following Monday 

 morning, between three and four a.m. About ten a.m. that day 

 my letter arrived, and on the first sheet, suie enough, there were 

 two blots. This had occurred by a little accident, and I had not 

 time to write a second letter. G. W.\tson. 



LETTERS RECEIVED AND SHORT ANSWERS. 



E. L. Gareett. Woe is me I that ever I admitted your Noah's 

 rainbow letter. It has brought me not only about a ream — more 

 or less — of controversial matter, but your own prodigious reply to 

 my short comments in your last. The discussion must close, or 

 else a whole number must be given up to it — which I must respect- 

 fully decline to suffer. No doubt, comets .Tre very nnmerous 

 (albeit Kepler was scarcely an authority on this point), and we 

 must have run into or through plenty, as we shall do over and over 

 again, without producing the slighte.st sensible effect upou the earth 

 — beyond, perhaps, the apparition of a rather finer shower of shoot- 

 ing stars than usual. You should know more of the rudiments of 

 Logic than to assume that aqueous comets must be held to exist 

 until the contrary is proved. It would be quite as rational for me 

 to assert that the inhabitants of Venus cat jiork chops on a Friday, 

 as you cannot prove the contrary ! A. B. C, and R. Laing. — As 

 you will see from the preceding reply, the Noah's Rainbow and 

 Flood discussion ia closed. — F. W. Hexkel. I am sorry that my 

 decision to stop the Flood discussion excludes your really valuable 

 and important communication.— .A. Goodall. Your letter to the 

 publishers was considerably deiayed, thanks to its being addreaaed 

 to the Editor. — J. You must read Knowledge mth very little 

 attention, or you would know that theology is rigidly excluded from 

 its columns, and your own letter is a theological one pure and 

 simple. Acquire some rudimentary idea of what evidence is from 

 Mill's " Logic." or any similar work. — H. P. Deane and S. Mackie. 

 Thanks ; but I was many miles away from South Kensington on 

 July 2. — D. M. Y'our letter on " Tricycle Tracks " is very much 

 longer than the interest of its subject warrants. — Peotea. The 

 black disc of A'enus herself forming the base of the shadow cone, 

 how can you possibh" expect to see her shadow in any degree 

 " sideways" during her transit either at ingress or egress? Why, 

 the whole diameter of the sun is little more than half a degree. 

 Besides, upon irhat do you suppose the shadow would be projected, 

 even were we looking (as it were) edgeways at the whole arrange- 

 ment ? The latest edition of my " Tr.insit3 of Venus " was pub- 

 lished in 1878. I know of no other work covering the same ground. 

 My lectures are always announced in Knowledge, but I have 

 ceased for the present to deliver them. — E. Pinfold and W. Coombee. 

 Y'our letters have been needlessly delayed owing to your omission 

 to address " the Chess Editor." — Depreciation. Mr. Wyman re- 

 grets that he has received no actuarial training. — T. Murray. The 

 *' P. D." theory is Pure Drivel, neither more nor less. Pray do not 

 waste your time in sending me expositions of it ; they one and all 

 find their way immediately into the waste-paper basket. — T. Maxn. 

 "Polyglot's" article could not have been written to enable all his 

 readers to dispense with the services of a patent agent, inasmuch 

 as it points out that in complicated cases such aid is indispensable. 

 Can you refer me to the number and page of Knowledge in which 

 the statement to which you refer occurs ? — Harold Rowntree. 

 Sorry that the compositor should have taken liberties with your 

 name. Thanks for your description of the approximate trisection 

 of an angle ; but, as in the ease of ninetenn-twentieths of the com- 

 munications I receive, it is too long for insertion. — James Cram. I 

 have received a " magic cube " from this gentleman which is really 

 a marvel, and worthy of notice by all interested in what may be 

 called numerical tricks. — A. G. Puller. I am away from my books 

 where I write, but will look up my authority on my return. — W. 

 Common. The change, t/made at all, would be made in longitude 

 180° ; but it is not a practical matter, as all civilised races may be 

 held to be comprised between auch hmits of longitude, that while 

 it is noon at Greenwich on, say, July 11, it is 3h. 40m. a.m. of the 



