with 41 instead of 17, we should have found the follow ing primes: 41, 43, 47, 53, 61, 71, &c., which fails at the 40th and 41st terms, which are respec tively the composite numbers 41 x 41, 41 x 43. If to the arithmetical progression 2, 6, 10, 14, 18, 22, &c., we commence with 29, and add as before, we shall find the series of primes, 29, 31, 37, 47, 61, 79, 101, &c., which holds good until we reach the 28th term, which is the composite number 59 × 29, after which it again gives primes. XVII. When a number terminates on the right with a zero, it is a whole number of tens, and is therefore divisible by 2 as well as by 5. When a number terminates with 5, it must be divsible by 5, since the tens are divisible by 5, and the units being 5, are also divisible by 5. When the number expressed by the two right-hand figures of any number is divisible by 4, the whole number is divisible by 4. For the number is composed of a whole number of hundreds and the number expressed by the two right-hand figures. The hundreds are always divisible by 4, and, by supposition, the part expressed by the two right-hand figures is also divisible by 4, consequently the whole number is divisible by 4. When the number expressed by the three right-hand figures of any number is divisible by 8, the whole num 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. No. of Primes. dec. + Primes. 3 P 7 C 11 No. I places. 822 413 276 839 419 853 213 1 163 81 P367 366 P593 592 P 823 37 C607 202 613 51 397 99 C617 88 P 857 856 631 315 P 863 862 643 107 C 881 440 P 29 28 C197 98 C409 204 41 43 21 227 113 450 936 P 61 60 C241 30 C457 152 C677 338 C 929 464 5 P487 486 83 41 271 P 97 96 C281 28 P499 498 C101 4 283 141 P503 502 C739 246 P 983 982 523 261 P743 742 991 495 C103 34 C293 146 P509 508 P149 148 C349 116 P571 570 C809 202 1039 519 |