SINGULAR PROPERTY OF THE FIGURE 9.

5. Every number will divide by 9, when the sum of its digits is divisible by 9.

For, take any number, as 78534; this number is, by the nature of decimal arithmetic, the same as 70000+ 8000+500+30+4.

... 76534-9999 × 7+999 x8+99x5+9x3+(7+8+ 5+3+4.)

Now, since each expression, 9999 x 7, 999 x 8, 99 × 5, and 9x3, is divisible by 9, it follows that the first number, 78534, will be divisible by 9 when the sum of its digits (7+8+5+3+4) is.

Hence, it follows that any number being diminished by the sum of its digits, will become divisible by 9.

Also, any number divided by 9, will leave the same remainder as the sum of its digits when divided by 9.

The above properties belong to the digit 3, as well as to that of 9, since 3 is a divisor of 9. No other digit has such properties.

NOTE.

These singular properties of the digit 9, have been made use of by many authors for proving the work of the four fundamental rules of arithmetic.

PRIME NUMBERS.

6. No even number can, with the single exception of the number 2, be a prime, since all even numbers are divisible by 2. It is also evident that there are many odd numbers which are not primes. If we write in order

the natural series of odd numbers, we discover that every third term, counting from 3, is divisible by 3; every fifth term, counting from 5, is divisible by 5; every seventh term, counting from 7, is divisible by 7, and so on.

Commencing at 3, under every third term, I have placed a small figure,,, to denote that the term under which it is placed is divisible by 3. Under every fifth term, counting from 5, I have, in like manner, placed a small indicating that the corresponding term is divisible by 5. I have proceeded in the same way for the higher primes. Now it is evident that all the terms, under which there are no small figures found, are primes.

59

We may also remark, that the numbers expressed by the small figures are the different prime factors of the numbers under which they are placed.

1, 3, 5, 7, 93, 11, 13, 153.5, 17, 19, 213.7, 23, 255, 273, 29, 31, 333.11, 355.7, 37, 393.13, 41, 43, 453.51 47, 49, 51.17, 53, 555.11, 573.19, 59, 61, 633.7, 655.13, 67, 693.23, 71, 73, 753.59 777.11, 79, 813, 83, 873-29, 89, 917.139 933-31) 955-19, 97, 993-117

855.179

&c.

In the above operation, we have found the primes only which are less than 100; but this process may be extended as far as we wish. This method of finding the successive primes was employed by Eratosthenes, who inscribed the series of odd numbers upon parchment, then cutting out such numbers as he found to be composite, his parchment with its holes resembled somewhat a sieve; hence, this method is called Eratosthenes' Sieve.

The number 2, although an even number, must be regarded as coming under our definition of a prime, since the only number which will divide it is itself.

TABLE OF PRIME NUMBERS.

1157 373 607 2163 379 613 3167 383 617 5 173 389 619 7179 397 631 11 181 401 641 13 191 409 643 17 193 419 647 19 197 421 653 23 199 431 659 29 211 433 661 31 223 439 673 37 227 443 677 41 229 449 683 43 233 457 691 47 239 461 701 53 241 463 709 59 251 467 719 61 257 479 727 67 263 487 733

8571103 1399 1637 1949 2239 2531 859 1109 1409 1657 1951 2243 2539 863 11171423 1663 1973 2251 2543 877 1123 1427 1667 1979 2267 2549 881 1129 1429 1669 1987 2269 2551 883 1151 1433 1693 1993 2273 2557 887 1153 1439 1697 1997 2281 2579 907 1163 1447 1699 1999 2287 2591 911 1171 1451 1709 2003 2293 2593 919 1181 1453 1721 2011 2297 2609 929 1187 1459 1723 2017 2309 2617 937 1193 1471 1733 2027 2311 2621 941 1201 1481 1741 2029 2333 2633 947 1213 1483 1747 2039 2339 2647 953 1217 1487 1753 2053 2341 2657 967 1223 1489 1759 2063 2347 2659 971 1229 1493 1777 2069 2351 2663 977 1231 1499 1783 2081 2357 2671 983 1237 1511 1787 2083 2371 2677 991 1249 1523 1789 2087 2377 2683 71 269 491 739 997 1259 1531 1801 2089 2381 2687 73 271 499 743 1009 1277 1543 1811 2099 2383 2689 79 277 503 751 1013 1279 1549 1823 2111 2389 2693 83 281 509 757 1019 1283 1553 1831 2113 2393 2699 89 283 521 761 1021 1289 1559 1847 2129 2399 2707 97 293 523 769 1031 1291 1567 1861 2131 2411 2711 101 307 541 773 1033 1297 1571 1867 2137 2417 2713 103 311 547 787 1039 1301 1579 1871 2141 2423 2719 107 313 557 797 1049 1303 1583 1873 2143 2437 2729 109 317 563 809 1051 1307 1597 1877 2153 2441 2731 113 331 569 811 1061 1319 1601 1879 2161 2447 2741 127 337 571 821 1063 1321 1607 1889 2179 2459 2749 131 347 577 823 1069 1327 1609 1901 2203 2467 2753 137 349 587 827 1087 1361 1613 1907 2207 2473 2767 139 353 593 829 1091 1367 1619 1913 2213 2477 2777 149 359 599 839 1093 1373 1621 1931 2221 2503 2789 151 367 601 853 1097 1381 1627 1933 2237 2521 2791

The preceding table contains all the primes which are not greater than 2791.

All prime numbers, except 2, are odd, and, therefore, terminate with an odd digit. Any number which ends with 5, is divisible by 5; hence it follows that all primes, except 2 and 5, must end with one of the figures, 1, 3, 7, or 9.

When it is required to determine whether a given number is a prime, we first notice the terminating figure; if it is different from 1, 3, 7, or 9, the number is a composite; but if it terminate with one of the above digits, we must endeavor to divide it by some one of the primes, as found in the table, commencing with 3. There is no necessity of trying 2, for 2 will divide only the even numbers. If we proceed to try all the successive primes of the table until we reach a prime which is not less than the square root of the number, without finding a divisor, we may conclude with certainty that the number is a prime.

The reason why we need not try any primes greater than the square root of the number, is drawn from the following consideration: If a composite number is resolved into two factors, one of which is less than the square root of the number, the other must be greater than the square root.

The square of the last prime given in our table is 7789681; hence, this table is sufficiently extended to enable us to determine whether any number not exceeding 7789681 is a prime. It is obvious that numbers may be proposed which would require by this method very great labor to determine whether they are primes, still this is the only sure and general method as yet discovered.

Tables have been calculated, giving not only all the primes up to 3036000, but also the least prime factor of the composite numbers up to the same extent.

Our table is of sufficient extent to enable us to work all ordinary examples.

Any prime number, except 2 and 3, when divided by 6, must have a remainder of 1 or 5; for all prime numbers are odd, and any odd number when divided by an even number, must leave an odd number for remainder. Hence, any odd number divided by 6, must give 1, 3, or 5, for a remainder; if the remainder is 3, the number must have been divisible by 3, since the divisor and remainder are each divisible by 3. Hence, the remainder. found by dividing a prime by 6, is 1 or 5. Therefore, by either adding one or subtracting one from any prime number greater than 3, it becomes divisible by 6.

7. Every number is either a prime number, or composed of prime factors.

For, all numbers which are not prime are composite, and can, therefore, be separated into two or more factors; and, if these factors are not prime, they can again be separated into other factors, and thus the decomposition can be continued until all the factors are prime.

Hence, to resolve any composite number into its prime factors, we have this

RULE.

Divide the number by any prime number which will divide it without any remainder; then divide the quotient in the same way, and so continue until a quotient is