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5X9+6+3+5+7)+9. But 6X999+3X99 +5X9 is evidently divisible by 9; therefore if 6357 be divided by 9, it will leave the same remainder as 6+3+5 to 7-9. The same will be found true of any other number whatever.

Obs. 1. This property of the number 9 affords an ingenious method of proving each of the fundamental rules. (Arts. 90, 123.) The same property belongs to the number 3; for, 3 is a measure of 9, and will therefore be contained an exact number of times in any number of 9s. But it belongs to no other digit.

2. The preceding is not a necessary but an incidental property of the number 9. It arises from the law of increase in the decimal notation. If the radit of the system were 8, it would belong to 7; if the radix were 12, it would belong to 11; and universally, it belongs to the number that is one less than the radix of the system of notation.

16. If the number 9 is multiplied by any single figure or digit, the sum of the figures composing the product, will make 9. Thus, 9 X4=36, and 3+6=9.

17. If we take any two numbers whatever ; then one of them, or their sum, or their difference, is divisible by 3. Thus, take 11 and 17; though neither of the numbers themselves, nor their sum is divisible by 3, yet their difference is, for it is 6.

18. Any number divided by 11, will leave the same remainder, as the sum of its alternate digits in the even places reckoning from the right, taken from the sum- of its alternate digits in the odd places, increased by 11 if necessary. Take any number, as 38405603, and mark the alternate fig

Now the sum of those marked, viz: 8+0+6+3=17. The sum of the others, viz: 3+4+5+0=12. And 17—12=5, the remainder sought. That is, 38405603 divided by 11, will leave remainder.

Again, take 5847362, the sum of the marked figures is 14; the sum of those not marked is 21. Now 21 taken from 25, (=14+11,) leaves 4, the remainder sought.

19. Every composite number may be resolved into prime factors. For, since a composite number is produced by multiplying two or more factors together, (Art. 160. Def. 3,) it may evidently be resolved into those factors; and if these factors themselves are composite, they also may be resolved into other factors, and thus the analysis may be continued, until all the factors are prime numbers.

20. The least divisor of every number is a prime number. For, every whole number is either prime, or composite ; (Art. 180.

ures,

Def. 2 ;) but a composite number, we have just seen, can be resolved into prime factors ; consequently, the least divisor of every number must be a prime number.

21. Every prime number except 2, if increased or diminished by 1, is divisible by 4. See table of prime numbers, next page.

22. Every prime number except 2 and 3, if increased or diminished by 1, is divisible by 6.

23. Every prime number, except 2 and 5, is contained without a remainder, in the number expressed in the common notation by as many 9s as there are units, less one, in the prime number itself.* Thus, 3 is a measure of 99; 7 of 999,999; and 13 of 999,999, 999,999.

24. Every prime number, except 2, 3, and 5, is a measure of the number expressed in common notation, by as many ls as there are units, less one, in the prime number. Thus, 7 is a measure of 111,111; and 13 of 111,111,111,111.

25. All prime numbers except 2, are odd; and consequently terminate with an odd digit. (Art. 160. Def. 4.)

Note.-1. It must not be inferred from this that all odd numbers are prime. (Art. 160. Def.6. Obs.)

2. It is plain that any number terminating with 5, can be divided by 5 without a remainder. Hence,

26. All prime numbers, except 2 and 5, must terminate with 1, 3, 7, or 9; all other numbers are composite.

161. a. To find the prime numbers in any series of numbers.

Write in their proper order all the odd numbers contained in the series. Then reckoning from 3, place a point over every third number in the series ; reckoning from 5, place a point over every fifth number ; reckoning from 7, place a point over every seventh number, and so on. The numbers remaining without points, together with the number 2, are the primes required.

Take the series of numbers up to 40, thus, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39; then adding the number 2, the primes are 1, 2, 3, 5, 7, 11, 13, &c.

Nole.—This method of excluding the numbers which are not prime from a series, was invented by Eratosthenes, and is therefore called Eratosthenes' Sieve.

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* Théorie des Nombres, par M. Legendre.

TABLE OF PRIME NUMBERS FROM 1 TO 3413.

1

1/173/4091659 941122315111811121292423 27413079 2179 419 661 9471229 1523 1823 2131 2437 27493083 3 181421 673 953 1231 153118312137 2441 2753 3089 5191 431 677 96712371543 1847 21412447 2767 3109 7193 433 683 9711249 1549 18612143 24592777 3119 11197 439 691 977 1259 1553 1867 2153 2467 2789 3121 13 199 443701 983 1277 1559 1871 2161 2473 27913137 17 211 449 709 991 1279 1567187321792477 2797 3163 19 223 457 719) 997 1283 1571 1877 2203 2503 2801 3167 23 227 461727.1009 1289 1579 1879 2207 2521 2803 3169 29 229463 733 1013 1291 1583 188922132531 2819 3181 31233 467 739 1019 1297 1597190122212539 2833 3187 37 239 479 743 10211301 1601 1907 2237 2543 28373191 41 241 487 751 1031 1303 16071913 2239 2549 2843 3203 43 251 4917571033 13071609 1931 22432551 2851 3209 47 257 499 761 1039 1319161319332251 2557 28573217 53 263 503 769 1049 13211619 1949 2267 2579 28613221 59269 509 773 1051132716211951 2269 2591 2879 3229 61 271 521 7871061136116271973 2273 2593 28873251 67277 523 797 1063136716371979 2281 2609 2897 3253 71281 541 809 10691373 1657 1987228726172903 3257 73 283 547 8111087 138116631993 2293 2621 2909 3259 79 293 557 8211091 1399 1667 1997 2297 2633 2917 3271 83 307 563 823 1093 1409 1669 19992309 2647 292713299 89 311 569 82710971423 1693200323112657 29393301 97 313 571829 110314271697 2011233326592953 3307 101 317 577 839 1109 1429 1699 2017 2339 2663 2957 3313 103 331587 85311171433 1709 2027 2341 267129633319 107 337 593 857 11231439 1721 2029 2347 2677 29693323 109 347 599 859 1129 1447172320392351 2683 2971 3329 113 349 601 863 1151 14511733 2053 2357 2687 29993331 127 353 607 877 1153 1453 1741 2063 2371 2689 3001 3343 131 359 613 8811163 1459 1747 20692377 2693 30113347 137 367 617 883 11711471 1753 2081 23812699 3019 3359 139373619 88711811481175920832383 2707 3023 3361 149 379 631 9071187 1483 1777 2087 23892711 30373371 151 383 641 9111193 1487 1783 2089 2393 2713 30413373 157389 643 91912011489 1787 2099 2399 2719 3049 3389 163 397 647 929 12131493 1789211124112729 3061 3391 167 401653937121711499 180121132417 2731 3067 3407

DIFFERENT SCALES OF NOTATION. 162. A number expressed in the decimal notation, may be changed to any required scale of notation in the following manner.

Divide the given number by the radix of the required scale continually, till the quotient is less than the radix; then annex to the last quotient the several remainders in a retrograde order, placing

ciphers where there is no remainder, and the result will be the number in the scale required. (Arts. 43, 44.)

Ex. 1. Express 429 in the quinary scale of notation,

Explanation.—By Dividing the given number 5)429 by 5, it is evidently distributed into 85 parts, 5) 85—4 each of which is equal to 5, with 4 remainder. 5) 1740 Dividing again by 5, these parts are distributed

3-2 into 17 other parts, each of which is equal to 5

Ans. 3204 times 5, and the remainder is nothing. Dividing by 5 the third time, the parts last found are again distributed into 3 other parts, each of which is equal to 5 times 5 into 5, with 2 remainder. Thus, the given number is resolved into 3X5X5X5+2X5X5+ 0X5+4, or 3204, which is the answer required. 2. Change 7854 from the decimal to the binary scale.

Ans. 1111010101110. 3. Change 7854 from the decimal to the ternary scale.

Ans. 101202220. 4. Change 7854 from the decimal to the quaternary scale.

Ans. 1322232. 5. Change 7854 from the decimal to the quinary scale.

Ans. 222404. 6. Change 7854 from the decimal to the senary scale.

Ans. 100210. 7. Change 7854 from the decimal to the octary scale.

Ans. 17256. 8. Change 7854 from the decimal to the nonary

scale.

Ans. 11686. 9. Change 7854 from the decimal to the duodecimal scale,

Ans. 4666,

10. Change 35261 from the decimal to the quaternary scale. 11. Change 643175 from the decimal to the octary scale. 12. Change 175683 from the decimal to the septenary scale. 13. Change 534610 from the decimal to the octary scale. 14. Change 841568 from the decimal to the nonary scale. 15. Change 592835 from the decimal to the duodecimal scale. Note.-Since every scale requires as inany characters as there are units in the radix, we will denote 10 by t, and 11 by e. Ans. 2470 t e.

163. To change a number expressed in any given scale of notation, to the decimal scale.

Multiply the left hand figure by the given radix, and to the product add the next figure ; then multiply this sum by the radix again, and to this product add the next figure ; thus continue the operation till all the figures in the given number have been employed, and the last product will be the number in the decimal scale. 16. Change 3204 from the quinary to the decimal scale.

Operation. Explanation.-Multiplying the left hand figure 3204 by 5, the given radix, evidently reduces it to the 5 next lower order; for in the quinary scale, 5 in 17 an inferior order make one in the next superior 5 order. For the same reason, multiplying this 85 sum by 5 again, reduces it to the next lower 5 order, &c.

429 Ans. Obs. This and the preceding operations are the same in principle, as reducing compound numbers from one denomination to another. 17. Change 1322232 from the quaternary to the decimal scale.

Ans. 7854. 18. Change 2546571 from the octary to the decimal scale. 19. Change 34120521 from the senary to the decimal scale. 20. Change 145620314 from the septenary to the decimal scale. 21. Change 834107621 from the nonary to the decimal scale. 22. Change 403130021 from the quinary to the decimal scale, 23. Change 704400316 from the octary to the decimal scale. 24. Change 903124106 from the duodecimal to the decimal scale.

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