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17. If 4 casks of vinegar contain 63gal. 3qt., what are the contents of one cask? What are the contents of 37 casks?

*Ans. 589gal. 2qt. 1 pt. 2gi. 18. When 5yd. 3qr. Ina. of cloth cost $ 4, how much cloth can be bought for $1 ? How much for $ 36 ?

Ans. 52yd. lqr. Ina. 19. If 11T. 3cwt. 2qr. of hay be sufficient to keep 4 horses 7°1 months, how much will keep 1 horse the same time? How much 23 horses ?

Ans. 64T. 5cwt. 12lb. 8oz. 20. If 12 men can dig a certain ditch in 286 days 4h. 33m., how long will it require 1 man to do the same labor? How long 72 men?

Ans. 47 days 16h. 45m. 30sec. 21. If 27yd. lqr. of cloth be required to make 21 coats, how many yards will be required to make 11 coats ?

22. If a train of cars move at the rate of 174m. 26rd in 7 hours, how far will it move in 1 hour ? How far in 10 hours ?

Ans. 248m. 5fur. 20rd. 23. If 4 cases of shoes, containing 60 pairs each, cost $ 192, what will 1 pair cost ? What will 25 cases cost ?

24. When 3A. 2R. 20rd. of land will buy 4 hogsheads of molasses, how much land will buy 1 hogshead ? How much 30 hogsheads?

Ans. 27A. OR. 30rd. 25. If a man can travel 20deg. 49m. 5fur. 35rd. 5yd. 3in. in 9 weeks, how far would he travel in 1 week? How far in 90 weeks?

Ans. 207deg. 13m. lfur. 25rd. 5yd.

PROPERTIES OF NUMBERS.

DEFINITIONS.

162. An integer is a whole number; as 1, 7, 16. All whole numbers are either prime or composite.

163. A prime number is a number which can be exactly divided only by itself and 1; as 1, 3, 5, 7, 11.

A composite number is a number which can be exactly divided by some number besides itself and 1; as 6, 9, 14, 18.

164. A factor of a number is such a number as will, by being taken an entire number of times, produce it; as, 3 is a factor of 9, and 4 a factor of 16.

165. A prime factor of a number is a prime number that will exactly divide it; thus the prime factors of 10 are the prime numbers 1, 2, and 5.

Note. – Unity or 1 is not generally regarded as a prime factor, since multiplying or dividing any number by 1 does not alter its value. It therefore will be omitted when speaking of the prime factors of numbers.

A composite factor of a number is a composite number that will exactly divide it; thus, 6 and 8 are composite factors of 48.

166. Numbers are prime to each other when they have no factor in common; thus, 4, 9, and 23 are prime to each other.

167. An aliquot part of a number is such a part as will exactly divide it; as, 1, 3, and 5 are aliquot parts of 15.

Note. — The aliquot parts of a number include all its factors, prime and composite.

An aliquant part of a number is such a part as will not exactly divide it; as 2, 4, 5, 7, and 8 are aliquant parts of 9.

168. The reciprocal of a number is the quotient arising from dividing 1 by the number; thus, the reciprocal of 2 is t.

169. The power of a number is the product obtained by taking the number a certain number of times as a factor; thus 25 is a power of 5.

NOTE. — When the number is taken once. it is called its first power; when taken twice, as a factor, the product is called its second power; and so on. The second power of a number is sometimes termed its square, and the third

power, its cube.

170. The exponent of

power is a figure written at the right of a number, and a little above it, to show how many times it is taken as a factor ; thus, in the expression 4”, the exponent is the 2, and the whole is read 4 second power; and in 73, it is the 3, and the whole read 7 third power.

Note. - The first power of a number being always the number itself, its exponent is not expressed.

PROPERTIES OF PRIME NUMBERS.

171. No direct process of detecting prime numbers has been discovered.

NotE. – A few facts, such as are given below, if kept in mind, will aid somewhat in ascertaining whether a number is prime or not.

172. The only even prime number is 2 ; since all other even numbers, as 4, 6, 8, and 10, it is evident, can be exactly divided by 2, and therefore must be composite.

173. The only prime number having 5 for a unit or righthand figure is 5 ; since every other whole number thus terminating, as 15, 25, 35, and 45, can be exactly divided by 5, and therefore must be composite.

174. Every prime number, except 2 and 5, must have 1, 3, 7, or 9 for the right-hand figure; since all other numbers are composite.

175. Every prime number above 3, when divided by 6, must leave 1 or 5 for a remainder; since every prime number above 3 is either 1 greater or 1 less than 6, or some exact number of times 6.

176. In a series of odd numbers written in their proper or natural order, if beginning with 3 every THIRD number, with 5 every FIFTH, with 7 every SEVENTH, be cancelled, as composite, the remaining numbers, with 2, will be the prime numbers of the natural series. Thus, in the series 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, every third number from the 3, every fifth from the 5, every seventh from the 7, every ninth from the 9, and so on, being cancelled, the remaining numbers, with 2, are all the prime numbers under 50.

NOTE 1. In the series, every third number from the 3 contains that number as a factor; every fifth number from the 5, that number as a factor; and

so on.

NOTE 2 The whole number of prime numbers from 1 to 100,000 is 9,593. Although all of these, except 2 and 5, end in 1, 3, 7, or 9, there are, within the same range, no less than 30,409 composite numbers terminating with some one of the same figures.

177. All the prime numbers not larger than 4057 are included in the following

TABLE OF PRIME NUMBERS.

563 569

577

31

659

1 233 557
2 239
3 241
5 251

571 7 257 11 263 587 13 269 593 17 271 599 19 277 601 23 281 607 29 283 613 293

617 37 307

619 41 311 631 43 313 641 47 317 643 53 331 647 59 337 653 61 347 67 349 661 71 1 353 673 73 359 677 79 367 683 83 373 691 89 379

701 97 383 709 101 389 719 103 397 727 107 401

733 109 409 739 113 419 743 127

421 751 131 431 757 137 433

761 139

439 769 149 443 773 151 449 787 157 457 797 163 461 809 167 463 811 173 467

821 179 479 823 181 487 827 191 491 829 193 499 839 197 503 853 199 509 857 211 521 859 223 523 863 227 541 1 877 229 547 881

883 887 907 911 919 929 937 941 947 953 967 971 977 983 991 997 1009 1013 1019 1021 1031 1033 1039 1049 1051 1061 1063 1069 1087 1091 1093 1097 1103 1109 1117 1123 1129 1151 1153 1163 1171 1181 1187 1193 1201 1213 1217 1223 1229 1231 1237

1613 1259 1619 1277 1621 1279 1627 1283 1637 1289 1657 1291 1663 1297 1667 1301 1669 1303 1693 1307 1697 1319 1699 1321 1709 1327 1721 1361 1723 1367 1733 1373 1741 1381 1747 1399 1753 1409 1759 1423

1777 1427 1783 1429 1787 1433

1789 1439 1801 1447 1811 1451 1823 1453 1831 1459 1847 1471 1861 1481 1867 1483 1871 1487 1873 1489 1877 1493 1879 1499 1889 1511 1901 1523 1907 1531 1913 1513 1931 1549 1933 1553 1949 1559

1951 1567 1973 1571 1979 1579 1987 1583 1993 1597 1997 1601 | 1999 1607 2003 1609 2011

2017 2027 2029 2039 2053 2063 2069 2081 2083 2087 2089 2099 2111 2113 2129 2131 2137 2141 2143 2153 2161 2179 2203 2207 2213 2221 2237 2239 2243 2251 2267 2269 2273 2281 2287 2293 2297 2309 2311 2333 2339 2341 2347 2351 2357 2371 2377 2381 2383 2389 2393

2399 2411 2417 2423 2437 2441 2447 2459 2467 2473 2477 2503 2521 2531 2539 2543 2549 2551 2557 2579 2591 2593 2609 2617 2621 2633 2647 2657 2659 2663 2671 2677 2683 2687 2689 2693 2699 2707 2211 2713 2719 2729 2731 2741 2749 2753 2767 2777 2789 2791 2797

2801 3253 | 3643 2803 3257 3659 2819 3259 3671 2833 3271 3673 2837 3299 3677 2843 3301 3691 2851 3307 3697 2857 3313 3701 2861 3319 3709 2879 3323 3719 2887 3329 3727 2897 3331

3733 2903 3343 3739 2909 3347 3761 2917 3359 3767 2927 3361 3769 2939 3371 3779 2953 3373 3793 2957 3389 3797 2963 3391 3803 2969 3407 3821 2971 3413 3823 2999 3433 3833 3001 3449

3847 3011 3457 3851 3019 3461 3853 3023 3463

3863 3037 3467

3877 3041 3469 3881 3049 3491 3889 3061 3499 3907 3067 3511 3911 3079 3517 3917 3083 3527 3919 3089 3529 3923 3109 3533 3929 3119 3539 3931 3121 3541 3943 3137 3547 3947 3163 3557 3967 3167 3559 3989 3169 3571 4001 3181 3581 4003 3187 3583 4007 3191 3593 4013 3203

3607 4019 3209 3:13 4021 3217 3617 4027 3221 3623 4049 3229 3631 4031 3251 3637 4037

FACTORING.

178. FACTORING is the process of resolving a quantity into its factors.

179. Every number that is not prime is composed of prime factors, since all numbers are either prime or composite ; and, if composite, can be separated into factors, which, if themselves composite, can be further separated into those that shall be prime.

OPERATION.

180. To resolve a composite number into its prime factors. Ex. 1. It is required to find the prime factors of 42.

Ans. 2, 3, 7. We divide by 2, the least prime number greater 2142 than 1, and obtain the quotient 21 ; and, since 21 is 3/21

a composite number, we divide this Sy 3, and obtain

for a quotient 7, which is a prime number. The 7

several divisors and the last quotient, all being prime,

constitute all the prime factors of 42, which, multiplied together, they equal. Hence

Divide the given number by any prime number that will exactly divide it, and the quotient, if a composite number, in the same manner; and so continue dividing, until a prime number is obtained for a quotient. The several divisors and the last quotient will be the prime factors required. NOTE 1.

- The composite factors of any number may be found by multiplying together two or more of its prime factors.

NOTE 2. Such prime factors as two or more numbers may have alike, are termed prime factors common to them; and these may be readily determinec after the numbers are resolved into their prime factors.

EXAMPLES. 2. What are the prime factors of 105 ? Ans. 3, 5, 7. 3. Resolve 220 into its prime factors. 4. What are the prime factors of 936 ?

Ans. 2, 2, 2, 3, 3, 13. 5. What are the prime factors of 1953 ? 6. Resolve 12462 into its prime factors. Ans. 2, 3, 31, 67. 7. Resolve 19987 into its prime factors. Ans. 11, 23, 79.

8. What are the prime factors common to 225, 435, and 540 ?

Ans. 3, 5.

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