§ XVI.

MISCELLANEOUS EXAMPLES IN MULTIPLICATION AND DIVISION OF COMPOUND NUMBERS.

1. BOUGHT 30 boxes of sugar, each containing Scwt. 3qr. 20lb., but having lost 68cwt. 2qr. Olb., I sold the remainder for 1£. 17s. 6d. per cwt.; what sum did I receive? Ans. 375£.

2. A company of 144 persons purchased a tract of land containing 11067A. 1R. 8p. John Smith, who was one of the company and owned an equal share with the others, sold his part of the land for 1s. 94d. per square rod; what sum did he receive? Ans. 1101£. 12s. 14d

3. The exact distance from Boston to the mouth of the Columbia River is 2644m. 3fur. 12rd. A man, starting from Boston, travelled 100 days, going 18m. 7fur. 32rd. each day; required his distance from the mouth of the Columbia at the end of that time. Ans. 746m. 7fur. 12rd. 4. James Bent was born July 4, 1798, at 3h. 17m. A. M.; how long had he lived Sept. 9, 1807, at 11h. 19m. P. M., reckoning 365 days for each year, excepting the leap year 1804, which has 366 days? Ans. 3353da. 20h. 2m.

5. The distance from Vera Cruz, in a straight line, to the city of Mexico, is 121m. 5fur. If a man set out from Vera Cruz to travel this distance, on the first day of January, 1848, which was Saturday, and travelled 3124rd. per day until the eleventh day of January, omitting, however, as in duty bound, to travel on the Lord's day, how far would he be from the city of Mexico on the morning of that day? Ans. 43m. 4fur. 8rd.

6. Bought 16 casks of potash, each containing 7cwt. 3qr. 18lb., at 5 cents per pound. I disposed of 9 casks at 6 cents per pound, and sold the remainder at 7 cents per pound; what did I gain? Ans. $182.39.

7. A merchant purchased in London 17 bales of cloth for 17£. 18s. 10d. per bale. He disposed of the cloth at Havana for sugar at 1£. 17s. 6d. per cwt. Now, if he purchased 144cwt. of sugar, what balance did he receive?

Ans. 35£. Os. 2d.

8. A and B commenced travelling, the same way, round an island 50 miles in circumference. A travels 17m. 4fur. 30rd. a day, and B travels 12m. 3fur. 20rd. a day; required how far they are apart at the end of 10 days.

Ans. 1m. 4fur. 20rd.

9. Bought 760 barrels of flour at $5.75 per barrel, which I paid for in iron at 2 cents per pound. The purchaser afterwards sold one half of the iron to an axe manufacturer; what quantity did he sell? Ans. 54T. 12cwt. 2qr

10. Bought 17 house-lots, each containing 44 perches, 200 square feet. From this purchase I sold 2A. 2R. 240ft., and the✨ remaining quantity I disposed of at 1s. 2d. per square foot; what amount did I receive for the last sale?

Ans. 5914£. 19s. 5d.

11. J. Spofford's farm is 100 rods square. From this he sold H. Spaulding a fine house-lot and garden, containing 5A. 3R. 17p., and to D. Fitts a farm 50rd. square, and to R. Thornton a farm containing 3000 square rods; what is the value of the remainder, at $1.75 per square rod? Ans. $6235.25. 12. Bought 78A. 3R. 30p. of land for $7000, and, having sold 10 house-lots, each 30rd. square, for $8.50 per square rod, I dispose of the remainder for 2 cents per square foot. much do I gain by my bargain?

How

Ans. $89265.35.

§ XVII.

PROPERTIES AND RELATIONS OF
NUMBERS.

ART. 112. AN INTEGER is a whole number; as 1, 6, 13.
All numbers are either odd or even.

An odd number is a number that cannot be divided by 2 without a remainder; thus, 3, 7, 11.

An even number is a number that can be divided by 2 without a remainder; thus, 4, 8, 12.

Numbers are also either prime or composite.

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

A composite number is a number which can be exactly divided other than by itself or 1; as 6, 9, 14.

Numbers are prime to each other when they have no factor in common; thus, 7 and 11 are prime to each other, as are, also, 4, 15, and 19.

QUESTIONS. Art. 112. What is an integer? What are all numbers? What is an odd number? What is an even number? What other distinctions of numbers are mentioned? What is a prime number? When are numbers prime to each other? What is a composite number?

All the prime numbers not larger than 1109 are included in the following

ART. 113. A prime factor of a number is a prime number that will exactly divide it; thus, the prime factors of 21 are the prime numbers 1, 3, and 7.

A composite factor of a number is a composite number that will exactly divide it; thus, the composite factors of 24 are the composite numbers 4 and 6.

NOTE 1.-Unity or 1 is not regarded as a material prime factor, since multiplying or dividing any number by 1 does not alter its value; it will be omitted when speaking of the prime factors of numbers.

NOTE 2.-There has been discovered no direct process by which prime numbers may be found. The following facts, however, if kept in mind, will aid in ascertaining whether a number is prime or not; and, if not prime, indicate one or more of its factors:

1. 2 is the only even prime number.

2. 2 is a factor of every even number.

3. 3 is a factor of every number the sum of whose digits 3 will exactly divide; thus, 15, 81, and 546, have each 3 as a factor.

4. 4 is a factor of every number whose two right-hand figures 4 will exactly divide; thus, 316, 532, and 1724, have each 4 as a factor.

5. 5 is the only prime number having 5 for a unit or right-hand figure.

QUESTIONS. Art. 113. What is a prime factor? What is a composite factor? How is unity or 1 regarded? Is there any direct process for de termining prime numbers? Which is the only even prime number? Of what numbers is 2 a factor? Of what numbers is 3 a factor? Of what numbers is 4 a factor?

6. 5 is a factor of every number whose right-hand figure is either 5 or 0; as, 15, 20, &c.

7. 6 is a factor of every even number that 3 will exactly divide; thus, 24, 108, and 360, have each 6 as a factor.

8. 7 is a factor of every number whose two right-hand figures are contained in the left-hand figure or figures exactly 3 times; thus, 602, 2107, and 3913, have each 7 as a factor.

9. 7 is a factor of every number occupying three or four places, when the two right-hand figures contain the left-hand figure or figures exactly 5 times; thus, 840, 945, and 1155, have each 7 as a factor.

10. 8 is a factor of every number whose three right-hand figures 8 will exactly divide; thus, 5072, 11240, and 17128, have each 8 as a factor. 11. 9 is a factor of every number the sum of whose digits 9 will exactly divide; thus, 27, 432, and 20304, have each 9 as a factor.

12. 10 is a factor of every number whose right-hand figure is 0; as, 20, 30, &c.

13. 7, 11 and 13, are factors of any number occupying four places in which two like figures have two ciphers between them; as, 3003, 4004, 9009, &c.

14. Every prime number, except 2 and 5, has 1, 3, 7, or 9, for the right-hand figure.

ART. 114. Method of finding the prime factors of numbers.

Ex. 1. It is required to find the prime factors of 24.

OPERATION.

2124

212

2 2

Ans. 2, 2, 2, 3.

We divide by 2, the least prime number greater than 1, and obtain the quotient 12. And since 12 is a composite number, we divide this also by 2, and obtain a quotient 6. We divide 6 by 2, and obtain 3 for a quotient, which is a prime number. The several divisors and the last quotient, all being prime, constitute all the prime factors of 24, which, multiplied together, 2×2×2×3=24 they equal.

6

3

RULE. Divide the given number by the least prime number, greater than 1, that will divide it, and the quotient, if a composite number, in the same manner; and 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.The composite factors of any number may be found by multiplying together two or more of its prime factors.

QUESTIONS. Of what numbers is 5 a factor? Of what numbers is 6 a factor? Of what numbers is 7 a factor? Of what numbers is 8 a factor? Of what numbers is 9 a factor? What is the right-hand figure of every prime number? What is the rule for finding the prime factors of numbers? How may the composite factors of numbers be found?

ART. 115. If the dividend and divisor are both divided by the same number, the quotient is not changed. Thus, if the dividend is 20 and the divisor 4, the quotient will be 5. Now, if we divide the dividend and divisor by some number, as 2, their proportion is not changed, and we obtain 10 and 2 respectively; and 1025, the same as the original quotient.

ART. 116. If a factor in any number is cancelled, the number is divided by that factor. Thus, if 15 is the dividend and 5 the divisor, the quotient will be 3. Now, since the divisor and quotient are the two factors, which, being multiplied together, produce the dividend (Art. 50), it is plain, if we cross out or cancel the factor 5, the remaining 3 is the quotient, and by the operation the dividend 15 has been divided by 5.

ART. 117. Cancellation is the method of shortening arithmetical operations by rejecting any factor or factors common to the divisor and dividend.

QUESTIONS. Art. 115. What is the effect on the quotient when the dividend and divisor are divided by the same number? What is the effect of cancelling a factor of any number? What is cancellation?