MISCELLANEOUS EXAMPLES.

1. Bought 30 boxes of sugar, each containing 8cwt. 3qr. 201b., 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, traveled 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 traveled 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 traveling, 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 ax 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. 23d. per square foot; what amount did I receive for the last sale?

Ans. 5914£ 19s. 5d.

From this he sold

11. J. Spofford's farm is 100 rods square. 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. do I gain by my bargain?

How much

Ans. $89265.35.

PROPERTIES OF NUMBERS.

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.

Integers are also either prime or composite numbers.

A Prime Number is a number which can be exactly divided by no integer except itself or 1; as, 1, 3, 5, 7.

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

Numbers are prime to each other when they have no factor (Art. 41) in common; thus, 7 and 11 are prime to each other, as are also 4, 15, and 19.

112. What is an integer?

What are all integers ? What is an odd number? 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

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 (Art. 41) 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 commonly regarded as a prime factor, since multiplying or dividing any number by I does not alter its value; it will be omitted when speaking of the prime factors of numbers.

NOTE 2. No direct process of finding prime numbers has been discovered. The following facts, however, will aid in ascertaining whether a number is prime or not; and, if not prime, will 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.

113. What is a prime factor? What is a composite factor? How is unity or 1 regarded? Is there any direct process for determining 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; ts, 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 occupying four places whose two righthand figures are contained in the left-hand figure or figures exactly 3 times; thus, 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 righthand figure.

114. To find the prime factors of numbers.

Ex. 1. Find the prime factors of 24.

OPERATION.

2124

212

2 6

3

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,

equal 2 X2 X2 X 3

RULE.

24.

- Divide the given number by any 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.

113. Of what numbers is 5 a factor?

7 a factor? Of what is 8 a factor?

Of what is 6 a factor? Of what is Of what is 9 a factor? What is the right-hand figure of every prime number?-114. The rule for finding the prime factors of numbers? How may the composite factors of numbers be found?

EXAMPLES FOR PRACTICE.

Ans. 2, 2, S, 3.

Ans. 2, 2, 2, 2, 3.

2. What are the prime factors of 36? 3. What are the prime factors of 48? 4. What are the prime factors of 56? 5. What are the prime factors of 144? 6. Find the prime factors of 3420? Ans. 2, 2, 3, 3, 5, 19. 7. What are the prime factors of 18500?

Ans. 2, 2, 2, 7. Ans. 2, 2, 2, 2, 3, 3.

Ans. 2, 2, 5, 5, 5, 37.

8. What are the prime factors of 19965?

Ans. 3, 5, 11, 11, 11.

9. What are the prime factors of 12496?

Ans. 2, 2, 2, 2, 11, 71.

10. What are the prime factors of 17199?

Ans. 3, 3, 3, 7, 7, 13.

11. What are the prime factors of 7800?

Ans. 2, 2, 2, 3, 5, 5, 13.

CANCELLATION.

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, we obtain 10 and 2 respectively; and 10 ÷ 2 = 5, the same as the original quotient.

Also, if the dividend and divisor are both multiplied by the same number, the quotient is not changed.

116. If a factor in any number is canceled, 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), 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.

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

115. What is the effect on the quotient when the dividend and divisor are divided by the same number? 116. The effect of canceling a factor of any number? -117. What is cancellation?