CHAPTER III.

COMPOSITE NUMBERS.-PRIME FACTORS.-COMMON MEASURE.

COMMON MULTIPLE.

COMPOSITE NUMBERS.

$ 64. A composite number is one which is the product of two factors, each greater than a unit.

Thus 4 is a composite number, being 2×2.

Is 6 a composite number? Is 7? Is 12? Is 19? Is 36? Is 45?

Decomposition of Numbers.

$65. Decomposing a number consists in resolving the number into its factors.

Thus 6 is decomposed when resolved into the factors 3 and 2.

Into what two factors may 15 be resolved? 21? 33 84 99? Into what three factors may 24 be resolved? 30? 70? 36? 100?

§ 66. In Division, the dividend is resolved into two factors, one of which is the divisor, and the other the quotient.

Taking 4 as a factor of 20, what is the other factor? 7 being one factor of 56, what is the other factor? 9 being one factor of 108, what is the other factor? 12 being one factor of 144, what is the other factor?

Any number whatever may be resolved into itself multiplied by a unit.

Thus 5 is 5X1; 7 is 7X1, &c.

Sign of Equality.

§ 67. The sign =, equal to, placed between two numbers, or numerical expressions, signifies that they are equal to each other.

Thus 12+8=4×5 signifies that the sum of 12 and 8 is equal to the product of 4 and 5; and is read 12 plus 8 is equal to 4 into 5.

Constant Product of Several Factors.

§ 68. The Product of several factors remains the same in whatever order the factors are multiplied together.

Take, for example, the product 2×3×5.

Since 2X33X2, we have 2×3×5=3×2×5;

and since 2X5=5×2, we have 3×2×5=3X5X2; and so on, there being six different ways in which the factors may be multiplied together.

Division by the Canceling of Factors.

§ 69. A Product is divided by either of its factors by canceling that factor; or by the product of any two or more of its factors, by canceling those factors.

For example, take 30=2×3×5.

If we divide it by 2, the quotient will be 3×5, or 15; and if we divide it by 2×3, or 6, the quotient will be 5 (§ 66).

If the product is to be divided by itself, all its factors must be canceled, and their place supplied by a unit; for a number is contained in itself once.

The cancellation of a number is denoted by a line drawn across it. Thus 2×3×5 denotes that the 2 in this product is canceled, which is equivalent to dividing said product by 2.

COMPOSITE MULTIPLIERS AND DIVISORS.

When a multiplier or divisor can be resolved into factors, each of which shall be a number not exceeding 12, or such number with Os annexed, it will sometimes shorten the operation to multiply or divide by means of such factors.

RULE IX.

$70. To multiply by means of FACTORS.

Resolve the multiplier into two or more factors; multiply by one of the factors, and the product thence arising by another factor; and so on, until all the factors are employed. The last product will be the one required.

To multiply 345 by 18.

EXAMPLE.

Resolving 18 into the factors 3 and 6, we have 345×3=1035; and 1035×6=6210. Then 345X18=345×3×6=6210 (§ 68).

EXERCISES.

In performing these exercises, use the factors of the multiplier. 1. Required the value of 147 shares of rail-road stock, at the rate of 96 dollars per share. Ans. 14112 dollars.

2. Allowing 63 gallons to fill a hogshead, how many gallons will be required to fill 183 hogsheads? Ans. 11529 gallons. 3. A planter sold 230 bales of cotton at an average of 32 dollars per bale. What sum did he receive for his cotton? Ans. 7360 dollars. 4. Allowing a ship to sail at the rate of 117 miles per day, how many miles would she sail in 108 days?

Ans. 12636 miles. 5. If 56 masons could build a certain wall in 310 days, in how many days could one mason build the same wall? Ans. 17360 days.

6. If 132 clerks can accomplish a certain amount of writing in 51 days, in what time could one clerk accomplish 3 times as great an amount of writing? Ans. 20196 days.

7. A gentleman purchased 42 bales of cotton cloth,—each bale containing 31 pieces, and each piece containing 36 yards. Required the number of yards that he purchased? Ans. 46872 yards.

8. A speculator bought a tract of land containing 1200 acres, at 72 dollars per acre; and afterwards sold one fifth of the tract at 96 dollars per acre. What did he gain on the part sold?

Ans. 5760 dollars.

RULE X.

§ 71. To divide by means of FACtors.

1. Resolve the divisor into two or more factors; divide by one of the factors, and the quotient thence resulting by another factor, and so on, until all the factors are employed. The last quotient will be the one required.

2. If a remainder occur in the first division, and in none succeeding it, it is the true remainder.

3. If a remainder occur in the second division, and in none succeeding it, multiply it by the first divisor, and to the product add the first remainder, if any, for the true remainder.

4. If three or more factors be used, multiply the last remainder by the preceding divisor, and to the product add the corresponding remainder, if any; multiply the sum by the next preceding divisor, adding as before; and so on, until the divisors are all included, for the true remainder.

EXAMPLE.

To divide 273 by 36.

4)273

9) 68 times, 1 over.

7 times, 5 over.

Quotient 7, true remainder 21; or quotient 731.

Resolving 36 into 4×9, we divide first by 4, and the quotient 68 thence resulting by 9, and obtain 7, the quotient required.

To find the true remainder, we multiply the second remainder 5, by the first divisor 4, and add the first remainder 1. Thus 5X4-20, and 1 makes 21.

The divisor 4 is only one ninth of the whole divisor 36; hence it is contained in the dividend 9 times as often as 36 is. The true quotient is then of that found for the divisor 4.

A remainder after the first division is so many units of the dividend. A remainder after the second division is so many units of the first quotient, and since the first quotient × the first divisor produces the dividend, a remainder of the first quotient the first divisor produces the corresponding remainder of the dividend. This remainder added to the first one, gives the true remainder of the dividend. ✈

EXERCISES.

In performing these exercises, use the factors of the divisor.

1. A hogshead of ale or beer contains 54 gallons; how many hogsheads then will be filled by 9479 gallons?

Ans. 175

hogsheads. 2. If 81 men take equal shares of 13846 dollars, how many dollars will be the share of each man? How will you find the answer to this question? How do you find of any number? Ans. 170 dollars.

3. Allowing a person to travel at the rate of 45 miles per hour, how long will he be in going 586 miles?

Ans. 13 hours.

4. Supposing 49 fat cattle to sell for 1975 dollars, what would be the average price for each? Ans. 40 dollars.

5. If one man can reap a field of hemp in 19 days, in what time ought 14 men to reap the same field?

Ans. 154.

6. In what time ought 72 men to accomplish the same amount of work that 9 men could do in 300 days?

Ans. 37

days.

7. If 77 cords of wood be purchased for 231 dollars, for what sum ought 521 cords to be bought at the same rate?

Ans. 1563 dollars.

8. Allowing 144 yards of cloth to sell for 864 dollars, what sum should be received for double the quantity of cloth, at double the price per yard? Ans. 3456 dollars. 9. A garrison of 140 men has provisions sufficient for 54 days. If 8 of the men depart, how long will the same provisions suffice the remainder of the garrison?

Ans. 573 days.

132

PRIME FACTORS.

§ 72. A prime number is one which cannot be resolved into two factors, each greater than a unit; thus 3 is a prime number. Is 5 a prime or a composite number? Is 8? Is 11? Is 15? Is 23? Name all the prime numbers, in succession from 1 to 23.

§ 73. Every composite number may be resolved into prime factors; that is, into factors each of which shall be a prime number.

For example, 30 may be at once resolved into 3×10; then resolving 10 into 2×5, we have 30=3×2×5; and 3, 2, and 5 are prime numbers.

What are the prime factors of 8? Of 63 ? Of 16 Of 21? Of 27?

Of 20? Of 24? Of 36? Of 33? Of 100?

Table of Prime Numbers above 23.

Of 1000?

This table, which might be extended without limit, may be useful to the pupil, by way of reference, in the application of subsequent Rules.

2911071199/311|421|541|647|769 883|1019|1129|1279 1427 1543 166318011951 31 109 211 313 431 547 653 773 887 1021 1151 1283 1429 1549 1667 1811 1973 37 118 223 317 43 557 659 787 907 1031 1153 1289 1433 1553 1669 1823 1979 41 127 227 331 439 565 661 797 911 10331163 1291 1439 1559 1693 1831 1987 43 131 229 337 445 569 673 809 919 1039 1171 1297 1447 1567 1697 1847 1993 47 137 233 347 449 571 677 11 929 1049 1181 1501 1451 1571 1699 1861 1997 53 139 239 349 457 577 683 821 937 1051 1187 1303 1453 1579 1709 1867 1999 59 149 241 353 461 587 691 823 941 1061 1193 1307 1459 1583 1721 1871 2003 61 151 251 359 463 595 701 827 947 1063 1201 1319 1471 1597 1723 1873 2011 67 157 257 367 467 599 709 829 953 1069 1213 1321 1481 1601 1733 1877 2017 71 163 263 373 479 601 719 839 967 10871217 1327 1483 1607 1741 1879 2027 73 167 269 379 487 607 727 653 971 1091 1223 1361 1487 1609 1747 1889 2029 79 173 271 383 491 613 733 857 977 1093 1229 1367 1489 1613 1753 1901 2039 83 179 277 389 499 617 739 859 983 1097 1231 1373 1493 1619 1759 1907 2053 89.181 281 397 503 619 743 863 991 1103 1237 1381 1499 1621 1777 1913 2063 97 191 2-3 401 509 631 751 871 997 1109 1249 1399 1511 1627 1783 1931 2069 101 193 293 409 521 641 757 877 1009 1117 1259 1 109 1523 1637 1787 1933 2081 103,197 307 419,523 64 761 881 10131123 1277 1423 1531 1657 1789 1949 2083