ber will be divisible by 8. For the number is composed of a whole number of thousands together with the num ber expressed by the three right-hand figures. The thousands are always divisible by 8, and, by supposition, the number expressed by the three right-hand figures is also divisible by 8, consequently the whole number is divisible by 8.

From the above, we are able to determine whether any given number is divisible by either of the numbers 2, 3, 4, 5, 6, 8, 9, 10, 11, 12.

Thus if we wish to know whether a number is divisible by 2, we must see whether it ends on the right with a zero or with the figure 2, in either case it is divisible by 2.

If we wish to know whether it is divisible by 3, we must see if the sum of its digits is divisible by 3, or see whether there is no excess of threes, if so, then the number is divisible by 3.

If we wish to know whether it is divisible by 4, we see whether the number expressed by its two right-hand figures is divisible by 4, if so, the number is divisible by 4.

If we wish to know whether it is divisible by 5, we see whether it ends on the right with a zero or a 5, if so, it is divisible by 5.

If we wish to know whether a number is divisible by 6, we first see if it is divisible by 2, we also see if it is divisible by 3, if so, then it is divisible by 6, since 6 equals 2 times 3.

If we wish to know whether it is divisible by 8, we see if the number expressed by its three right-hand figures is divisible by 8, if so, the number is divisible by 8.

If we wish to know whether a number is divisible by 9, we see whether the sum of its digits is divisible by 9, if so, the number is divisible by 9.

If we wish to know whether a number is divisible by 10, we notice whether the number end on the right with a zero, if so, the number is divisible by 10.

If we wish to know whether a number is divisible by 11, we see if the difference of the sum of the digits of the even places, and those of the odd places, is zero, or a multiple of 11, which may be most readily done by subtracting the first or left-hand digit from the second, and this result from the third, and the result now obtained from the fourth, and so on, as explained under V. of this Appendix. If then the difference is either zero, or a multiple of 11, the number is divisible by 11.

If we wish to know whether a number is divisible by 12, we see whether it is divisible by 3, we also see if it is divisible by 4, if so, it will be divisible by 3 times 4, or by 12.

There is no simple practical method of determining whether a number is divisible by 7. The best way, in this case, is to make the trial by the method of short division.

By the application of the above rules, which in practice will be found to be very simple, we may at once abbreviate or reduce vulgar fractions, when any of these factors are common to the numerator and denominator.

If the numerator and denominator are both primes, the fraction is in its lowest terms. Also, if either the numerator or denominator is a prime, and not a divisor of the other, the fraction is in its lowest terms.

When a decimal fraction is expressed by writing its

denominator, it becomes a vulgar fraction whose denominator is some power of 10, and therefore contains no prime factors except 2 and 5, hence if the numerator is not divisible by 2 or 5, the fraction is in its lowest terms.

LIII. We have seen that Complementary repetends, which include the Perfect repetends, arise from expressing the reciprocals of primes by the aid of decimals. All prime numbers will not give rise to complementary repetends, they will all give Simple repetends, except 2 and 5. The following table, shows the number of places of decimals in each period of the simple repetend, arising from the primes up to 1051, except the primes 2 and 5, whose reciprocals are accurately expressed by decimals. Those in the table marked with C give complementary repetends, and those marked with P are not only complementary repetends, but are also perfect repetends. It will be seen, by inspecting this table, that when the number of decimal places is one less than the prime, it is not only complementary but also perfect.

At first view the pupil might imagine the labor of forming this table to be exceedingly great, on account of the great number of places in some of the periods; this labor would truly be immense were we obliged to find these decimals by the usual method, but if we employ the process explained under Art. 45, the work is rapidly performed. Indeed the number of places in a period may be found even without actually finding the decimal figures, by a method which is rather simple, but which would require considerable space to explain.

3

1

C 11

C 13

163 81 P367 366 P 593 592 P 823 822 6 P167 166 C373 186 599 299 827 413 173 43 P379378 C601 300 C 829 276

839 419

2 6 P 179 178 P383 382 C607 202 P 17 16 P181 180 P 389 388 613 51 853 213 P 19 18 191 95 397 99 C617 88 P 857 856 P 23 22 P 193 192 C401 200 P619 618 C 859 26 P 29 28 C197 98 C409 204 31 15 199 99 P419 418 3 C211 30 C421|140 5 P223 222 43 21 227 113 P 47 46 P229 228

37

41

53 13 P233 232

P 59 58 239

631 315 P 863 862 C641 32 C 877 438 643 107 C 881 440 431 215 P647 646 883 441 P433 432 C653 326 P 887 886 907 151

439 219 P659 658

443 221 C661 220 C 911 450 7C449 32 C673 224 919 459

P 61 60 C241 30 C457152 C677 338 C 929 464

719 359 C 967 P727 726 P 971

936

952

322

970

67 33 C251 50 P461 460 683 341 P 937 71 35 P257 256 C463 154 C691 230 P 941 940 C 73 8 P263 262 467 233 P701 700 947 473 79 13 P269 268 479 239 P709 708 P 953 83 41 271 5 P487 486 C 89 44 277 69 P491 490 P 97 96 C281 28 P499 498 733 61 P 977 976 C101 4 283 141 P503 502 C739 246 P 983 982 C103 34 C293 146 P509 508 P743 742 991 495 107 53 307 153 C521 52 751 125 C 997 166 P109 108 311 155 523 261 757 27 1009 252 P113 112 P313 312 P541 540 C761 380 1013 253 C127 42 317 79 547 91 C769 192 P1019 1018 P131 130 C331 110 C557 278 C137 8 P337 336 563 281 C139 46 347 173 C569 284 P149 148 C349 116 P571 570 C809 202 1039 519

773 193 P1021 1020 787 393 1031 103 797 199 P1033 1032

151 75 C353 32 P577 576 P811 810 C1049 524 C157 78 359179 587 293 P821 820 |P1051 1050

We have shown that the reciprocal of all prime numbers, except 2 and 5, when expressed by decimals, give repetends whose number of places of figures cannot exceed the units, less one, contained in the prime number. When the number of places in the period is less than the units of the prime, after one has been subtracted, it must be a multiple of this number. So that if the process of decimating the reciprocal of any prime number, except 2 and 5, be carried to as many decimals, less one, as there are units in the prime, the remainder must of necessity be a unit. That is, if we divide the power of 10, which is denoted by an exponent which is one less than our prime divisor, the remainder will be 1. Hence, if from this power we subtract 1, the remainder will be exactly divisible by this prime. When 10 is raised to any power, it will consist of 1 placed to the left of as many zeros as there are units in the exponent, and if we subtract 1, the result will be a number denoted by as many nines as there was zeros. Hence, any prime divisor, except 2 and 5, is exactly contained in a number which consists of as many successive nines as there are units, less one, in the prime.

As examples, 3 is a divisor of 99; 7 is a divisor of 999999; 11 is a divisor of 9999999999; 13 is a divisor of 999999999999; and so on.

As each of these expressions is divisible by 9, which contains only the prime 3, their quotient, which will consist of a succession of 1s, must be divisible by the same primes as before, excepting in the case of the prime 3, which is a factor of 9. That is, any prime divisor, except 2, 3, and 5, is exactly contained in a number which