46 



KNOWLEDGE. 



[January 1, 1890. 



digits 0. 1, 6, .j only, and consequently no integer ending 

 in 2, 8, 4, 7, H, 9 can have its fourth root an integer. It 

 is useful to note that the fourth power of any prime (except 



2 and 5) ends in iniit;/. Thus J'* — y/* is divisible by 240, 

 at least when /'and /< are prime numbers > 5. 



Any power of -5, 6, or 10 or number ending in 5, (>. 

 ends in .5, 6 or 0. Also, since 5 = 1-4-2-, wo have by 

 I'^uler's Theorem 5" = the sum of two integral squares 

 where n is anv integer whatever. 



Thus 52=r;#-f 42'; 5'=22+ll- = -)--l-10-, &c. &c. 



It is remarkable that the tiflli power of any number ends 

 in the same integer as the number itself does, thus l'=l ; 

 2==;-i2: 35=243, &c. 



This fact can be utilised :— The fifth i-oot of 17.210,368 

 is an integer. Determine it by inspection. Pomting off 

 as usual, we have two figures in the root, and the imits' 

 digits must be 8 for the reason given. And since 3= is 

 > 172 we must have 2 in the tens' place. Thus we get 

 28 for the fifth root. 



Again, Ball inhis Histuri/ nf Mntliciiiallis, p. 2G2, says :— 

 " There is no integral solution of the equation ,r" + i/"=;:" 

 if 11 is any integer > 2. Euler proved it when h = 3, and 

 Lagrange gives a proof when w=4. It appears to be 

 generally true. The riddle therefore awaits a solu- 

 tion." 



Now, if .r» + //==;5, and all are integers, we must also 

 have .r+i/=:, or (10»'-f-), and the equation may be 

 solved. 



If we proceed with the sixth, seventh, Ac. powers of 

 numbers, we shall find they are but repetitions of the 

 squares, cubes, &c. So that we have the first, fifth, ninth 

 --4«- 3rd power of an integer ending in the same digit as 

 the number itself. 



Similarly X-'-"-'" cannot end ui the digits 2, 3, 7, 8, but 

 X'"-^ may end in kiu/ digit, and .V" can only end in the 

 digits 0, i, 6, r,. 



('irciiliitinij 1 >rrii)i(ih. 



There is evidently a close connection between Prime 

 Numbers and Circulating Decimals. The rules relating 

 to the conversion of vulgar fractions into decimals are 

 generally ignored in the ordmary arithmetics on accoimt 

 of their number and intricacy. In order to determine the 

 periodicity of any vulgar fraction l/.V we can reduce the 

 rules to two. If the vulgar fraction be of the Form l/.V = 

 1 2' .5" the decimal tcnnintiirs after y/ figures from the 

 decimal point, y/ being the greater of the numbers .r and //. 

 In all other cases it will mnr, c.'/. let x — 3, 1/ = 2, then 

 l/.V ^ l/2» S"- = 1/200 = -005. 



If IjX be the vulgar fraction w^here .V ends in anv one 

 of the digits 1, 3, 7, or 9 then if (10" - 1) jX or (.V" + X 

 - 1) /.V be an integer // will give the periodicity. Thus, 

 (■.,/., if V= 13 we have (10'' -1) /13 or (4" + 12) ,13 an 

 integer when y< = 6. Thus the number of figures in the 

 period of yV will be 6. 



Since the vulgar fi-action is always supposed to be 

 brought down to its lowest terms, a numerator would not 

 all'ect the periodicity. The rule as usually given can only 

 toll us that the inimber of figures in the period is equal 

 either to (^(.V) or to a sub multiple of ^(.V) where <t>(X) 

 denotes the totitives of V, i.r. the number of numbers less 

 than .V and prime to it. 



It is a curious fact that when A' is a prime, the figures 

 of the period of 1 .V are divisible by 9. When the 

 period is cirii it is also divisible by 11, and when the 

 period contains Sn figures by 37, &c. This w^ill be 

 enlarged upon in a future number. Thus c.i/. -g\ = 

 •627 and -I- 2 -f 7 = 9 ; ^\ = -62439 and -f 2 -)- 4 -f 



3 + 9 = IS ; J^ = -076923 = 27 ; 7V = -0126582278481 

 = 54 ; also .L ; J^, &.C. &c. 



Of course this is true generally when the period con- 

 tains X - 1 figures. We may also note such fractions as 



,ij — -015873, where the second half is complementary to 

 the first. Thus 3-1-5 = 8; 7-1-1=8; 8-|-0=8; similarly 



Jy= -008547 gives us 555 ; ^j give us G6G, &c. &c. 



I am now in a position to show one advantage that the 

 method of division given in the November issue has over 

 the ordinary method. 



Find the periodicity of ^j. I set four boys simul- 

 taneously to work. Two do the first and third jiortions 

 simultaneously by the ordinary method of division, and 

 two others work the second and fourth portions by the 

 new method till the figures meet or overlap. I then test 

 the result by the nines. 



Thus A's work would be to divide 1-000 -I- by 01 = fifteen 

 figures. 



B's work would be to iindtiplii by 6 and point off till he 

 has fifteen figures, as explained in the November issue, 

 thus:— 1000-0 + 



6 



Thus we have + 7540 = last four figures of 2nd portion. 

 C"s work will be to di^•ide 60-000 -I- by 61 to fifteen 

 figures, and D will multiply by 6 as follows by the new 

 method :— ^ 999-9 



54 

 94-5 

 30 

 90-4 

 24 

 ~7^ 

 Thus we have -i-2459 = last four figures of period. 

 The test is— -i- 7540 



+ 2459 

 + 9999 

 We can thus set four boys to work one division question 

 simultaneously, and it is interesting to note that we are 

 able to obtain the middle figures of a quotient, and to 

 work the question each way till the whole result is 



A 

 obtained. The full result is J^ = -016393442622950, 



B C ■ I) 



819672131147510; 983606557377019, 180327868852459 

 = 9999+, &3., if A and C and B and D be added 

 together. This result may be obtained in a much easier 

 manner, as will be exnlained in another pajjer. 



I will now describe a method of writing down mechani- 

 cally the figures of the period of 1/A' where .N' is a 

 prime. 



Primes > 5 end in the figures 1, 3, 7, or 9 only. It is 

 easy to deduce from this that the last figure of the period 

 of 1/A' must be 9, 3, 7, or 1, respectively. Let us take 

 JLj for example. The hixt figure, according to the rule 

 just given, is 1. All we have to do now' is to multiply 

 that 1 by 2 ; then multiply the product 2 by 2 ; then the 

 product 4 by 2, and so on till the figures irj/mt. Thus the 

 fuU period is Jg= • 652631578947368421. 



Again, write down mechanically the period of g'g. 

 Multiply the last figure, which is 1, by 3. Then the pro- 

 duct 3 by 3, and so on as before. Thus we get J^ = 

 : 634482758620()89655172413793i. 



It is remarkable how interesting the dreariest part of 

 arithmetic becomes when scientifically investigated. 



